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Patterns, Functions, and Algebra DAY 1 PFA5.6 You can use a graph to show the relationship between two quantities. Points of the graph are located by using ordered pairs of numbers. The first number of an ordered pair tells the number of spaces you move to the right or left. (x) The second number tells the number of spaces to move up or down. (y) Example: Joe walk at the rate of 3 kilometers per hour. The table below shows the relationship between d, the distance he walks, and t, the time it takes him to walk this distance. What does the graph look like? Time t (h) 0 1 2 3 Distance d (km) 0 3 6 9 Ordered Pair (0,0) (x, 3) (2,6) (3,y) Function Rule: d =3 x t Use the data in the table to graph this relation. Step 1: Label the axes. Write time (h) to label the horizontal axis. Write Distance (km) to label the vertical axis. Step 2: Plot the points. Step 3: Connect the points. As distance increases, time The graph shows a line that What does the graph look like? Day 2 You probably recognize the relationships between variables and their affect on each other in real life. For example: 1) The more your puppy eats, the more or less he weighs? 2) As the temperature decreases in the winter, do you need to wear more layers of clothing or less? Example: Using the data from the table, describe the relationship between the amount of days left before summer break and the number of passes sold to the Zoo. Days left Before Summer Break 30 25 20 15 10 5 * As one variable variable # of Zoo Passes Sold 100 125 175 300 400 525 , the other . Is the rate of change constant? Day 3 A sequence is a set of numbers arranged in a pattern. 2, 5, 8, 11, 14,…. Example: Esther wrote the number sequence below on a piece of paper. She asked her friend Marsha to find the next two terms in the sequence. What are the next two terms? 1, 3, 6, 10, 15, … Day 4: A function is a relation in which one quantity is dependent on the other. A function can be described in words or by using an equation where variables are used to represent each quantity. Example: Julia used toothpicks to make the triangles below. Describe the toothpick function in words and with an equation. Based on the function, how many toothpicks are needed to make 8 triangles? Step 1: Complete the function table to help. Number of triangles (t) Number of toothpicks (p) 1 2 3 4 3 5 X W Step 2: Determine how you could change each number in the top row to get the number in the bottom row. X2 +1 Step 3: Describe the function. The number of toothpicks is equal to twice the number of triangles plus . p= xt+ Step 4: Use the function to find p when t = 8. Day 5 Step 1: Study the diagram. Look for a pattern. Step 2: Apply the pattern. How many blocks will there be on the seventh step? The total number of blocks is equal to the sum of . Step 3: Check your answer. Draw a seven-step staircase. Count all the blocks. How many were needed? Day 6 Remember an unknown number may be represented by a letter or other symbol called a variable. An algebraic expression is an expression that contains one or more variables. Example: Jalise had some pencils. Mary Kate gave her 4 more pencils. Write an expression to determine how many pencils Jalise has now. Use p for pencils. Since we do not know the amount she started or ended with, we can write an expression, but not an equation. P+4 An equation is a mathematical sentence that states that two expressions are equal. Each number in a sequence is called a term. You can find any term in a sequence of numbers if you have an algebraic expression for the nth term of the sequence. Step 1: Identify the rule that the pattern follows. Make a table that lists the given terms and the value of each term. 1st Term Value of Term 1 2nd 3rd 3 6 4th 5th 10 15 Step 2: Note how the values change from term to term. First term:1 Second Term: 3=1+2 Third Term: 6=3+3 Fourth Term: 10=6+4 Fifth Term: 15=10+5 The number being added to each term increases by . Step 3: Use the pattern rule to find the sixth and seventh terms. Sixth: Seventh: Day 7 Inequality: a mathematical sentence that states that two expressions are not equal! Two symbols are commonly used: < means less than > means greater than Inequalities are solved the same as equations, ONLY, the solution (answer) may be more than one number. Example: Gabby had a greater number of spelling words to learn than her sister who is in second grade. Gabby’s sister had 15 words on her list. Write an inequality to represent the number of spelling words (N) Gabby could have. Solution: Gabby’s sister had 15. Gabby could have any number greater than 15. So: N >15 N > 15, not = to 15, because the word problem stated that Gabby had more than her sister and her sister had 15. The variable N represents any number greater than 15 (16, 17, 18, 19, 20, etc) Example: Put in the correct symbol (<, >) below. 20 + 15 45 + 5