Patterns and Expressions Lesson 1-1 Algebra 2 Patterns and Expressions Lesson 1-1

Goals Goal Rubric To identify and describe patterns. Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

Essential Question Big Idea: Variable How do variables help you model real-world situations? Students will use an expression to model the nth term of a pattern. Students will use variables to represent unknown quantities in real-world situations.

Vocabulary Constant Variable quantity Variable Numerical expression Algebraic expression

Patterns What is the pattern of geometric figures below? How can you identify a pattern? Look for the same type of change between consecutive figures. Each figure in the pattern is created by rotating the previous figure 90˚ clockwise.

Patterns What does the eighth figure in the pattern look like? Because each figure is rotated 90˚, there are four figures in the pattern. Therefore, the eighth figure will be the same as the fourth figure.

Patterns Patterns can be represented using; Diagrams (geometric patterns, graphs, etc.) Words Numbers Algebraic expressions

Geometric Patterns Look at the figures from left to right. What is the pattern? Look for the same type of change between consecutive figures. The pattern shows regular polygons with the number of sides increasing by one. Hexagon has 6 sides. Triangle has 3 sides. Square has 4 sides. Pentagon has 5 sides.

Geometric Patterns What would the next figure in the pattern look like? The last figure in the pattern has six sides, so the next figure would have seven sides, a heptagon.

Your Turn: Look at the figures from left to right. What is the pattern? Draw the next figure in the pattern? Answer: The pattern shows a red center square and a yellow square added to each side with the number of squares per side increasing by one.

Definitions Constant – a quantity whose value does not change. Examples: 8, -23, π Variable Quantity – a quantity that can have values that vary. Examples: distance, time, cost

Definitions Variable – a symbol, usually a letter, that represents one or more numbers. Examples: n, x Numerical Expression – a mathematical phrase that contains numbers and operation symbols. Examples: 3 + 5, (8 – 2) + 5 Algebraic Expression – a mathematical phrase that contains one or more variables. Examples: 3n + 5, (8x – 2) + 5n

Question? What is the only difference between algebraic expressions and numerical expressions? Answer: An algebraic expression contains one or more variables, while a numerical expression contains no variables.

The nth Term By "the nth term" of a pattern we mean an algebraic expression that will allow us to calculate the term that is in the nth position (any position) of the pattern. For example consider the pattern: 1st term 2nd term 3rd term 4th term …nth term 3 9 27 81 … 3n

Algebraic Patterns Use the pattern to determine how many toothpicks are in the 20th figure?

Algebraic Patterns You can use a table to find the pattern that relates the figure number (term) to the number of toothpicks? 1st term 2nd 3rd 4 4 4 4 4 4 Term (input) Process # Toothpicks (output) 1 2 3 … 20 ⨯ 1 4 ⨯ 2 8 ⨯ 12 3 20 ⨯ 4 80 Pattern: To get the number of toothpicks (output), multiply the term (input) by 4.

Algebraic Patterns What is an algebraic expression for the number of toothpicks in the nth figure (the nth term)? Use the pattern from the table. The expression is formed by multiplying the term number, n, by 4. There are 4n toothpicks in the nth figure. Term (input) Process # Toothpicks (output) 1 4⨯1 4 2 4⨯2 8 3 4⨯3 12 … n 4⨯n 4n

Two Stage Algebraic Expressions Algebraic Patterns Two Stage Algebraic Expressions Some patterns are not as straight forward and you may have to use two stages. This means that after multiplying your term number, you may need to add or subtract a number to reach your answer.

Example The pattern

Pattern table Term 1 2 3 4 5 # of Squares 7 9

The Process or Rule Term 1 2 3 4 5 # of Squares 7 9 Notice that the pattern is adding 2 to each figure. And almost matches the multiples of 2 so lets see how that works.

The First Stage Term 1 2 3 4 5 # of Squares 7 9 Term× 2 6 8 10 We are not quite there, what do we need to do to reach the number of squares?

The Second Stage Term 1 2 3 4 5 # of Squares 7 9 Term×2 6 8 10 Subtract 1 We need to subtract one after we have multiplied.

The Algebraic Expression The algebraic expression is: 2 × the term number - 1 Remember to change to algebra we use variables: the term number is always "n“ 2n - 1

Example: Geometric patterns can be represented numerically and generalized algebraically.  

Example Continued Let’s create a table to see the relationship between each build and the number of blocks…

Example Continued Let’s create a table to see the relationship between each build and the number of blocks… Term # Description Process # of blocks

Example Continued Let’s create a table to see the relationship between each build and the number of blocks… Term # Description Process # of blocks 1 1 row of 2 plus 1 1(2)+1 3 Term #1

Example Continued Let’s create a table to see the relationship between each build and the number of blocks… Term # Description Process # of blocks 1 1 row of 2 plus 1 1(2)+1 3 2 2 rows of 2 plus 1 2(2)+1 5 Term #2 Term #1

Example Continued Let’s create a table to see the relationship between each build and the number of blocks… Term # Description Process # of blocks 1 1 row of 2 plus 1 1(2)+1 3 2 2 rows of 2 plus 1 2(2)+1 5 3 rows of 2 plus 1 3(2)+1 7 Term #3 Term #2 Term #1

Example Continued Let’s create a table to see the relationship between each build and the number of blocks… Term # Description Process # of blocks 1 1 row of 2 plus 1 1(2)+1 3 2 2 rows of 2 plus 1 2(2)+1 5 3 rows of 2 plus 1 3(2)+1 7 The number changing in each process is the number of rows. Term #3 Term #2 Term #1

Example Continued Let’s create a table to see the relationship between each build and the number of blocks… Term # Description Process # of blocks 1 1 row of 2 plus 1 1(2)+1 3 2 2 rows of 2 plus 1 2(2)+1 5 3 rows of 2 plus 1 3(2)+1 7 The number changing in each process is the number of rows. Term #3 So the rows are our variable (n)… And the nth term is: 2n + 1 Term #2 Term #1

Your Turn: How many squares are in the 25th figure in this pattern? Use a table of values. What is the algebraic expression for the nth term?

Solution: Term (input) Process # Squares (output) 1 2(1) + 2 4 2 2(2) + 2 6 3 2(3) + 2 8 … 25 2(25) + 2 52 The pattern is to multiply the term number by 2 and then add 2. So, there are 2(25) + 2 = 52 squares in the 25th figure. The algebraic expression for the nth term is: 2n + 2

Graphed Patterns You want to set up an aquarium and need to determine what size tank to buy. The graph shows tank sizes using a rule that relates the capacity of the tank to the combined lengths of the fish it can hold. If you want five 2-in. red fish, four 1-in. blue fish, and a 3-in. green fish, which is the smallest capacity tank you can buy: 15-gal, 20-gal, or 25-gal? Use a table to find the answer.

Graphed Patterns First, calculate the combined fish length. Wanted five 2-in. red fish, four 1-in. blue fish, and one 3-in. green fish. First, calculate the combined fish length. (5)(2) + (4)(1) + 3 = 17 in. Now, how can you use the graph to find the size tank needed? You can use the graph to make a table and find a pattern relating combined fish length (input) to tank capacity (output).

Graphed Patterns How do you make a table from the graph? First, chose points that have whole number coordinates, like (0, 2), (5, 7), and (10, 12). Then make the table using the input and output values shown in the ordered pairs. Length (input) Process Cap. (output) 2 5 7 10 12

Graphed Patterns Next, use the process column of the table to find the pattern relating length (input, x-coordinate) to capacity (output, y-coordinate). Therefore, the pattern is; output = input + 2 Using variables, let L = combined length of fish and C = capacity of the tank, then C = L + 2 Length (input) Process Cap. (output) 2 5 7 10 12 0 + 2 5 + 2 10 + 2

Graphed Patterns Finally, we need to answer the question “which is the smallest capacity tank you can buy: 15-gal, 20-gal, or 25-gal”. Using C = L + 2 and L = 17, C = 19 or the tank capacity we need is 19-gal. Therefore, the smallest capacity tank you can buy is 20-gal.

Your Turn: The graph shows the total cost of platys at the aquarium shop. How much do six platys cost? Use a table and find an equation relating number platys to total cost.

Solution Output = Input • 2 Using variables, let P = number platys and C = total cost, then C = 2P. For six platys, P = 6 and C = 2 • 6 C = 12 Six platys cost $12.

Number Patterns What is the pattern? Look for the same type of change between consecutive numbers or terms. Find the next term in each pattern. 1, 1, 2, 3, 5, 8, 13, ____ 1, 4, 9, 16, 25, 36, ____ O, T, T, F, F, S, S, E, N, ____ 21 49 T

Your Turn: Find the next term in each pattern. 1, 4, 3, 16, 5, 36, 7, ____ 6, 8, 5, 10, 3, 14, 1, ____ B, 0, C, 2, D, 0, E, 3, F, 3, G, ____ 64 18 2

Assignment Section 1-1, Pg 7 – 10; #1 – 7 all, 8 – 48 even.