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Lines of Numbers – Identify Patterns Reason Proportionally

Teacher Notes

In Lines of Numbers, students examine repeating patterns, predict the n^{\text{th}} number in the pattern, and figure out the sum of the first n numbers in the pattern for various values of n.

Solutions

1. \!\\{\;}  \quad \text{a}.\ \text{The}\ 25^{\text{th}}\ \text{number is}\ 1: \!\\ {\;}  \qquad \ 25 \div 4\ \text{is}\ 6\ \text{with one left over and}\ 4 \times 6 = 24.\ \text{So the}\ 24^{\text{th}}\ \text{number is}\ 4,\ \text{the last number in} \!\\{\;}  \qquad \ \text{the set. The}\ 25^{\text{th}}\ \text{number is}\ 1,\ \text{the first number in the set}. \!\\{\;}  \quad \text{b}.\  \text{The}\ 75^{\text{th}}\ \text{number is}\ 3: \!\\{\;}  \qquad \ 75 \div 4 = 18\ \text{with} \ 3 \ \text{left over and}\ 4 \times 18 = 72.\ \text{So the}\ 72^{\text{nd}}\ \text{number is}\ 4.\!\\{\;}  \qquad \ \text{The}\ 73^{\text{rd}}\ \text{number is}\ 1,\ \text{the}\ 74^{\text{th}}\ \text{is}\ 2, \ \text{and the}\ 75^{\text{th}}\ \text{is}\ 3. \!\\{\;}  \quad \text{c}.\ \text{The sum of the first}\ 75\ \text{numbers is}\ 186: \!\\{\;}  \qquad \ \text{The sum of one set of}\ 1,\ 2,\ 3, \ \text{and}\ 4\ \text{is}\ 10.\ \text{From the answer to question b,\ we know that} \!\\{\;}  \qquad \ 18\ \text{sets of four numbers ends with}\ 72.\ \text{The sum of the first 72 numbers is} \!\\{\;}  \qquad \ 10 \times 18,\ \text{or}\ 180.\ \text{The three left over numbers are}\ 1,\ 2,\ \text{and}\ 3,\ \text{and their sum is}\ 6.\ \text{So the} \!\\{\;}  \qquad \ \text{sum of the first}\ 75\ \text{numbers in the pattern is}\ 180 + 6,\ \text{or}\ 186.\!\\2. \!\\{\;}  \quad \text{a}.\ \text{The}\ 40^{\text{th}}\ \text{number is}\ 2: \!\\ {\;}  \qquad 40 \div 3\ \text{is}\ 13\ \text{with one left over and}\ 3 \times 13 = 39.\ \text{So the}\ 39^{\text{th}}\ \text{number is}\ 4,\ \text{the last number} \!\\{\;}  \qquad \text{on the set. The}\ 40^{\text{th}}\ \text{number is}\ 2,\ \text{the first number in the set}. \!\\{\;}  \quad \text{b}.\ \text{The}\ 110^{\text{th}}\ \text{number is}\ 3: \!\\ {\;}  \qquad 110 \div 3 = 36\ \text{with} \ 2 \ \text{left over and}\ 3 \times 36 = 108.\ \text{So the}\  108^{\text{th}}\ \text{number is}\ 4. \!\\ {\;}  \qquad \text{The}\ 109^{\text{th}}\ \text{number will be}\ 2,\ \text{and the}\ 110^{\text{th}}\ \text{is}\ 3. \!\\{\;}  \quad \text{c}.\ \text{The sum of the first}\ 110\ \text{numbers is}\ 329: \!\\ {\;}  \qquad \text{The sum of one set of}\ 2,\ 3,\ \text{and}\ 4\ \text{is}\ 9.\ \text{From the answer to question b,\ we know that}\ 36 \!\\{\;}  \qquad \text{sets of three numbers ends with}\ 108.\ \text{The sum of the first}\ 108\ \text{numbers is}\ 9 \!\\{\;}  \qquad \times 36,\ \text{or}\ 324.\ \text{The two left over numbers are}\ 2\ \text{and}\ 3,\ \text{and their sum is}\ 5.\ \text{So the sum of the} \!\\{\;}  \qquad \text{first}\ 110\ \text{numbers in the pattern is}\ 324 + 5,\ \text{or}\ 329.

 3. \!\\{\;}  \quad \text{a}.\ \text{The}\ 70^{\text{th}}\ \text{number is}\ 3: \!\\{\;}  \qquad \ 70 \div 4\ \text{is}\ 17\ \text{with two left over and}\ 4 \times 17 = 68.\ \text{So the}\ 68^{\text{th}}\ \text{number is}\ 7,\ \text{the last number}\!\\{\;}  \qquad \ \text{in the set. The}\ 69^{\text{th}}\ \text{number is}\ 1,\ \text{and the}\ 70^{\text{th}}\ \text{is}\ 3. \!\\{\;}  \quad \text{b}.\ \text{The}\ 175^{\text{th}}\ \text{number is}\ 5: \!\\{\;}  \qquad \ 175 \div 4 = 43\ \text{with} \ 3 \ \text{left over and}\ 4 \times 43 = 172.\ \text{So the}\ 172^{\text{nd}}\ \text{number is}\ 7. \!\\ {\;}  \qquad \ \text{The}\ 173^{\text{rd}}\ \text{number is}\ 1,\ \text{the}\ 174^{\text{th}}\ \text{is}\ 3,\ \text{and the}\ 175^{\text{th}}\ \text{is}\ 5. \!\\{\;}  \quad \text{c}.\ \text{The sum of the first}\ 175\ \text{numbers is}\ 697: \!\\ {\;}  \qquad \text{The sum of one set of}\ 1,\ 3,\ 5,\ \text{and}\ 7\ \text{is}\ 16.\ \text{From the answer to question b, we know that}\!\\{\;}  \qquad 43\ \text{sets of four numbers ends with}\ 172.\ \text{The sum of the first}\ 172\ \text{numbers is}\!\\{\;}  \qquad 16 \times 43,\ \text{or}\ 688.\ \text{The three left over numbers are}\ 1,\ 3,\ \text{and}\ 5,\ \text{and their sum is}\ 9.\ \text{So the} \!\\{\;}  \qquad \text{sum of the first}\ 175\ \text{numbers in the pattern is}\ 688 + 9,\ \text{or}\ 697.\!\\4. \!\\{\;}  \quad \text{a}.\ \text{The}\ 47^{\text{th}}\ \text{number is}\ 8: \!\\ {\;}  \qquad \ 47 \div 3\ \text{is}\ 15\ \text{with two left over and}\ 3 \times 15 = 45.\ \text{So the}\ 45^{\text{th}}\ \text{number is}\ 9,\ \text{the last number} \!\\{\;}  \qquad \ \text{in the set. The}\ 46^{\text{th}}\ \text{number is}\ 7,\ \text{the first number in the set, and the}\ 47^{\text{th}}\ \text{number is}\ 8. \!\\{\;}  \quad \text{b}.\ \text{The}\ 100^{\text{th}}\ \text{number is}\ 7: \!\\ {\;}  \qquad \ 100 \div 3 = 33\ \text{with} \ 1 \ \text{left over and}\ 3 \times 33 = 99.\ \text{So the}\ 99^{\text{th}}\ \text{number is}\ 9. \!\\{\;}  \qquad \ \text{The}\ 100^{\text{th}}\ \text{number is}\ 7. \!\\{\;}  \quad \text{c}.\ \text{The sum of the first}\ 100\ \text{numbers is}\ 809: \!\\ {\;}  \qquad \text{The sum of one set of}\ 7,\ 8,\ \text{and}\ 9\ \text{is}\ 24.\ \text{From the answer to question b, we know that}\ 33 \!\\{\;}  \qquad \text{sets of three numbers ends with}\ 99.\ \text{The sum of the first}\ 99\ \text{numbers is}\ 33 \!\\{\;}  \qquad \times 24,\ \text{or}\ 802.\ \text{The}\ 100^{\text{th}}\ \text{number is}\ 7.\ \text{So the sum of the first}\ 100\ \text{numbers in the pattern is}\!\\{\;}  \qquad 802 + 7,\ \text{or}\ 809.

5. \!\\{\;}  \quad \text{a}.\ \text{The}\ 27^{\text{th}}\ \text{number is}\ 6: \!\\{\;}  \qquad \ 27 \div 4\ \text{is}\ 6\ \text{with three left over and}\ 4 \times 6 = 24.\ \text{So the}\ 24^{\text{th}}\ \text{number is}\ 8,\ \text{the last number on} \!\\{\;}  \qquad \ \text{the set. The}\ 25^{\text{th}}\ \text{number is}\ 4,\ \text{the first number in the set, the}\ 26^{\text{th}}\ \text{number is}\ 4,\ \text{and the}\ 27^{\text{th}} \!\\{\;}  \qquad \ \text{number is}\ 6. \!\\{\;}  \quad \text{b}.\ \text{The}\ 78^{\text{th}}\ \text{number is}\ 4: \!\\{\;}  \qquad \ 78 \div 4 = 19\ \text{with} \ 2 \ \text{left over and}\ 4 \times 19 = 76.\ \text{So the}\ 76^{\text{th}}\ \text{number is}\ 8. \!\\{\;}  \qquad \ \text{The}\ 77^{\text{th}}\ \text{number is}\ 4,\ \text{and the}\ 78^{\text{th}}\ \text{is}\ 4. \!\\{\;}  \quad \text{c}.\ \text{The sum of the first}\ 78\ \text{numbers is}\ 426: \!\\{\;}  \qquad \text{The sum of one set of}\ 4,\ 4,\ 6,\ \text{and}\ 8\ \text{is}\ 22.\ \text{From the answer to question b, we know that} \!\\{\;}  \qquad 19\ \text{sets of four numbers ends with}\ 76.\ \text{The sum of the first} \ 76\ \text{numbers is} \!\\{\;}  \qquad 19 \times 22,\ \text{or}\ 418.\ \text{The two left over numbers are}\ 4\ \text{and}\ 4,\ \text{and their sum is}\ 8.\ \text{So the sum of} \!\\{\;}  \qquad \text{the first}\ 78\ \text{numbers in the pattern is}\ 418 + 8,\ \text{or}\ 426.

6. \!\\{\;}  \quad 1.\ \text{The}\ 70^{\text{th}}\ \text{number is}\ 3: \!\\{\;}  \qquad \ 70 \div 4\ \text{is}\ 17\ \text{with two left over and}\ 4 \times 17 = 68.\ \text{So the}\ 68^{\text{th}}\ \text{number is}\ 7,\ \text{the last number}\!\\{\;}  \qquad \ \text{in the set. The}\ 69^{\text{th}}\ \text{number is}\ 1,\ \text{and the}\ 70^{\text{th}}\ \text{is}\ 3. \!\\{\;}  \quad 2.\ \text{The}\ 175^{\text{th}}\ \text{number is}\ 5: \!\\ {\;}  \qquad \ 175 \div 4 = 43\ \text{with} \ 3 \ \text{left over and}\ 4 \times 43 = 172.\ \text{So the}\ 172^{\text{nd}}\ \text{number is}\ 7. \!\\ {\;}  \qquad \text{The}\ 173^{\text{rd}}\ \text{number is}\ 1,\ \text{the}\ 174^{\text{th}}\ \text{is}\ 3,\ \text{and the}\ 175^{\text{th}}\ \text{is}\ 5. \!\\{\;}  \quad 3.\ \text{The sum of the first}\ 175\ \text{numbers is}\ 697: \!\\ {\;}  \qquad \text{The sum of one set of}\ 1,\ 3,\ 5,\ \text{and}\ 7\ \text{is}\ 16.\ \text{From the answer to question 2, we know that} \!\\{\;}  \qquad 43\ \text{sets of four numbers ends with}\ 172.\ \text{The sum of the first}\ 172\ \text{numbers} \!\\{\;}  \qquad \text{is}\ 16 \times 43,\ \text{or}\ 688.\ \text{The three left over numbers are}\ 1,\ 3,\ \text{and}\ 5,\ \text{and their sum is}\ 9.\ \text{So the} \!\\{\;}  \qquad \text{sum of the first}\ 175\ \text{numbers in the pattern is}\ 688 + 9,\ \text{or}\ 697.

& \mathbf{Describe:} && \text{The number pattern keeps repeating the numbers}\ 3,\ 4,\ \text{and}\ 4. \\\\& \mathbf{My \ job:} && \text{Figure out the}\ 50^{\text{th}}\ \text{and}\ 200^{\text{th}}\ \text{numbers and the sum of the first}\ 200\ \text{numbers}. \\\\& \mathbf{Plan:} && \text{The numbers are in sets of three}:\ 3,\ 4,\ \text{and}\ 4. \ \text{For each question,} \\& && \bullet \text{Find how many sets of three numbers there are and how many numbers are left over}. \\& && \bullet \text{Then to find the}\ 50^{\text{th}}\ \text{and the}\ 200^{\text{th}}\ \text{numbers}. \\  & && \bullet \text{To find the sum of the first}\ 200\ \text{numbers,\ multiply the number of sets of three} \\& && \quad \text{numbers times the sum of the three numbers,\ and add the number left over numbers.} \\\\& \mathbf{Solve:} && \text{a}. \quad \text{The}\ 50^{\text{th}}\ \text{number is}\ 4: \\ & && \qquad 50 \div 3\ \text{is}\ 16\ \text{with} \ 2 \ \text{left over and}\ 3 \times 16 = 48. \\& && \qquad \text{So the}\ 48^{\text{th}}\ \text{number is}\ 4,\ \text{the last number in the set}.\\& && \qquad \text{The}\ 49^{\text{th}}\ \text{number will be}\ 3,\ \text{and the}\ 50^{\text{th}}\ \text{will be}\ 4.

& && \text{b}. \quad \text{The}\ 200^{\text{th}}\ \text{number is}\ 4: \\ & && \qquad 200 \div 3 = 66\ \text{with} \ 2 \ \text{left over and}\ 3 \times 66 = 198. \\ & && \qquad \text{So the}\ 198^{\text{th}}\ \text{number will be}\ 4.\ \text{The}\ 199^{\text{th}}\ \text{number will}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\& && \qquad \text{be}\ 3,\ \text{and the}\ 200^{\text{th}}\ \text{will be}\ 4.

& \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; && \text{c}.\quad \text{The sum of the first}\ 200\ \text{numbers is}\ 333: \\& \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; && \qquad 11\  \text{is the sum of one set of}\ 3,\ 4,\ 4 \\& \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; && \qquad \text{There are}\ 66\ \text{sets of three numbers ends with}\ 198.\\& \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; && \qquad 11 \times 66,\ \text{or}\ 726\ \text{is the sum of the first}\ 198\ \text{numbers}.\ 3 + 4,\ \text{or}\ 7\ \text{is the sum of the} \\& \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; && \qquad \text{two left over numbers}. \\& \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; && \qquad \text{So},\ 726 + 7,\ \text{or}\ 733\ \text{is the sum of the first}\ 200\ \text{numbers in the pattern}.

& \mathbf{Check:} \\& && \text{a}.\quad \text{The}\ 1^{\text{st}},\ 4^{\text{th}},\ 7^{\text{th}},\ \text{and so forth numbers are}\ 3'\text{s}.\ \text{All positions one more than a}\\& && \qquad \text{multiple of}\ 3\ \text{are}\ 3'\text{s}.\ \text{So the}\ 49^{\text{th}}\ \text{number is}\ 3\ \text{and the}\ 50^{\text{th}}\ \text{is}\ 4. \\& && \text{b}.\quad 199\ \text{is one more than a multiple of}\ 3,\ \text{so the}\ 199^{\text{th}}\ \text{number is}\ 3\ \text{and the}\ 200^{\text{th}}\ \text{is}\ 4. \\& && \text{c}.\quad \text{There are}\ 66\ \text{sets of} \ 3 \ \text{in}\ 200,\ \text{with two left over. The sum of the first}\\& && \qquad 66\ \text{sets of three numbers is}\ 11 \times 66,\ \text{or}\ 726,\ \text{and the sum of the two extra numbers} \\& && \qquad \text{is}\ 7,\ \text{so the sum is}\ 733.

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CK.MAT.ENG.SE.1.Algebra-Explorations-K-7.9.9

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