1. Multiple Relationships Within Number Patterns 5.OA.B.3

2. 04 + 4 8 12 + 4 16 + 4 02 + 2 4 6 8 (4,2)(8,4)(12,6)(16,8)(0,0) Here are two 0s.We will use two different repeating functions to generate two patterns.Here is the first pattern …… and here is the second pattern.The corresponding terms form ordered pairs, starting with (0,0) …(4,2) …(8,4) …(12,6) …… and (16,8). Multiple Relationships Within Number Patterns

3. 2 4 6 8 10 12 14 16 18 20 22 24 2 4 6 8 10 12 14 (4,2)(8,4)(12,6)(16,8)(0,0) Let’s use a coordinate plane to graph these ordered pairs.First, let’s graph ordered pair (0,0).(4,2)(8,4)(12,6)(16,8)All of these points share a single line. Multiple Relationships Within Number Patterns x-axis y-axis

4. 59 + 4 13 + 4 17 + 4 21 + 4 57 + 2 9 11 + 2 13 + 2 (9,7)(13,9)(17,11)(21,13)(5,5) Here are two 5s.Notice we are using the same two repeating functions to generate new patterns.Here is the first pattern …… and here is the second pattern.The corresponding terms form the ordered pairs (5,5) …(9,7) …(13,9) …(17,11) …… and (21,13). Multiple Relationships Within Number Patterns

5. 2 4 6 8 10 12 14 16 18 20 22 24 2 4 6 8 10 12 14 x-axis y-axis (9,7)(13,9)(17,11)(21,13)(5,5) This is the graph that we made earlier using the same two functions.First, let’s graph ordered pair (5,5).(9,7)(13,9)(17,11)(21,13)All of these new points share a new line.That line is parallel to the points produced earlier using the same set of functions. Multiple Relationships Within Number Patterns

6. 03 + 3 6 9 12 + 3 01 + 1 2 3 4 (3,1)(6,2)(9,3)(12,4)(0,0) Here are two 0s.We will use two new repeating functions to generate two new patterns.Here is the first pattern …… and here is the second pattern.The corresponding terms form ordered pairs, starting with (0,0) …(3,1) …(6,2) …(9,3) …… and (12,4).Let’s begin with new patterns and a new coordinate plane. Multiple Relationships Within Number Patterns

7. 2 4 6 8 10 12 14 16 18 20 22 24 2 4 6 8 10 12 14 x-axis y-axis (3,1)(6,2)(9,3)(12,4)(0,0) Let’s use a coordinate plane to graph these ordered pairs.First, let’s graph ordered pair (0,0).(3,1)(6,2)(9,3)(12,4)All of these points share a single line. Multiple Relationships Within Number Patterns

8. 58 + 3 11 + 3 14 + 3 17 + 3 56 + 1 7 8 9 (8,6)(11,7)(14,8)(17,9)(5,5) Here are two 5s.Notice we are using the same two repeating functions to generate new patterns.Here is the first pattern …… and here is the second pattern.The corresponding terms form the ordered pairs (5,5) …(8,6) …(11,7) …(14,8) …… and (17,9). Multiple Relationships Within Number Patterns

9. 2 4 6 8 10 12 14 16 18 20 22 24 2 4 6 8 10 12 14 x-axis y-axis (8,6)(11,7)(14,8)(17,9)(5,5) This is the graph that we made earlier using the same two functions.First, let’s graph ordered pair (5,5).(8,6)(11,7)(14,8)(17,9)All of these new points share a new line.That line is parallel to the points produced earlier using the same set of functions. Multiple Relationships Within Number Patterns

10. 1 + 3 8 × 2 210 312 414 516 618 + 7 + 8 + 9 + 10 + 11 + 12 What is the final outcome of this number pattern?If we begin with 2, what is the outcome?10What if we begin with 3?12If we begin with 4?14How about 5?16What if we begin with 6?Notice that 1 + 7 = 8.2 + 8 = 103 + 9 = 124 + 10 = 145 + 11 = 16And, 6 + 12 = 18.Let’s find additional connections between number patterns.818There are many patterns and number connections to be found in this table. Multiple Relationships Within Number Patterns

11. 33 46 59 612 715 818 + 0 + 2 + 4 + 6 + 8 + 10 Multiple Relationships Within Number Patterns – 2× 3 What is the final outcome of this number pattern?If we begin with 4, what is the outcome?6What if we begin with 5?9If we begin with 6?12How about 7?15What if we begin with 8?Notice that 3 + 0 = 3.4 + 2 = 65 + 4 = 96 + 6 = 127 + 8 = 15And, 8 + 10 = 1818There are many patterns and number connections to be found in this table.3

12. 523 628 733 838 943 1048 + 18 + 22 + 26 + 30 + 34 + 38 Multiple Relationships Within Number Patterns × 5– 2 What is the final outcome of this number pattern?If we begin with 6, what is the outcome?28What if we begin with 7?33If we begin with 8?38How about 9?43What if we begin with 10?Notice that 5 + 18 = 23.6 plus what number = 28?7 plus what number = 33?268 plus what number = 38? 3048There are many patterns and number connections to be found in this table.23229 plus what number = 43?34And, 10 plus what number = 48?38

13. 208 249 2810 3211 3612 4013 – 12 – 15 – 18 – 21 – 24 – 27 Multiple Relationships Within Number Patterns ÷ 4+ 3 What is the final outcome of this number pattern?If we begin with 24, what is the outcome?9What if we begin with 28?10If we begin with 32?11How about 36?12What if we begin with 40?Notice that 20 – 12 = 8.24 minus what number = 9?28 minus what number = 10?1832 minus what number = 11?2113There are many patterns and number connections to be found in this table.81536 minus what number = 12?24And, 40 minus what number = 13?27

14. Closing Question Multiple Relationships Within Number Patterns

15. 27 414 621 828 1035 12 + 5 + 10 + 15 + 20 + 25 + 3042 What is the final outcome of this number pattern?If we begin with 4, what is the outcome?14What if we begin with 6?21If we begin with 8?28Notice that 2 + 5 = 7.4 plus what number = 14?6 plus what number = 21?158 plus what number = 28?20Before we go further, let’s find a second pattern that we could use and check.710What number should we add to 10?25What is 10 + 25?35Here is the pattern of numbers that we will begin with this time.Let’s return to our first two functions to check our pattern.Is 35 the correct outcome?Yes, it is.And, is 42 the correct outcome?What number should we add to 12?30What is 12 + 30?42Yes, it is. Multiple Relationships Within Number Patterns × 7÷ 2