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Analyze patterns with two rules

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5509895
Two number patterns both start at \(2\). Pattern A rule: add \(3\). Pattern B rule: add \(5\). Write the first five terms of each pattern, including the starting term.

Hints

- Use the starting term as the first term in each pattern. - Apply the same add rule repeatedly within one pattern. - Keep the two patterns separate while generating their terms.

Solution

1. Pattern A starts at \(2\) and increases by \(3\): \(2, 5, 8, 11, 14\). 2. Pattern B starts at \(2\) and increases by \(5\): \(2, 7, 12, 17, 22\).

Answer

Pattern A: \(2, 5, 8, 11, 14\) Pattern B: \(2, 7, 12, 17, 22\)
5509905
Pattern A starts at \(4\) and follows the rule “add \(6\).” Pattern B starts at \(9\) and also follows the rule “add \(6\).” Complete the table and describe the relationship between corresponding terms. <table><tr><th>Position</th><th>Pattern A</th><th>Pattern B</th></tr><tr><td>1</td><td>\(4\)</td><td>\(9\)</td></tr><tr><td>2</td><td>\(10\)</td><td></td></tr><tr><td>3</td><td>\(16\)</td><td>\(21\)</td></tr><tr><td>4</td><td>\(22\)</td><td></td></tr><tr><td>5</td><td>\(28\)</td><td>\(33\)</td></tr></table>

Hints

- Extend Pattern B using its stated add rule before comparing the patterns. - Compare terms in the same position, not terms from different positions. - Look for a relationship that remains true across all five corresponding pairs.

Solution

1. Pattern B increases by \(6\), so its missing terms are \(15\) and \(27\). 2. Compare corresponding terms: \(9 - 4 = 5\), \(15 - 10 = 5\), \(21 - 16 = 5\), \(27 - 22 = 5\), and \(33 - 28 = 5\). 3. Each Pattern B term is \(5\) greater than the corresponding Pattern A term.

Answer

Missing Pattern B terms: \(15\) and \(27\). Relationship: each Pattern B term is \(5\) greater than the corresponding Pattern A term.
5509915
Pattern A starts at \(3\) and follows the rule “add \(3\).” Pattern B starts at \(6\) and follows the rule “add \(6\).” Complete the table and state how each Pattern B term is related to the corresponding Pattern A term. <table><tr><th>Position</th><th>Pattern A</th><th>Pattern B</th></tr><tr><td>1</td><td>\(3\)</td><td>\(6\)</td></tr><tr><td>2</td><td></td><td>\(12\)</td></tr><tr><td>3</td><td>\(9\)</td><td></td></tr><tr><td>4</td><td></td><td>\(24\)</td></tr><tr><td>5</td><td>\(15\)</td><td>\(30\)</td></tr></table>

Hints

- Extend each pattern using only its own add rule. - Compare terms that have the same position number. - Ask whether one corresponding term can be obtained by multiplying the other by the same number each time.

Solution

1. Extending Pattern A gives \(3, 6, 9, 12, 15\). Extending Pattern B gives \(6, 12, 18, 24, 30\). 2. The missing entries are \(6\), \(18\), and \(12\) in their respective cells. 3. In every position, the Pattern B term is twice the corresponding Pattern A term.

Answer

Completed Pattern A: \(3, 6, 9, 12, 15\) Completed Pattern B: \(6, 12, 18, 24, 30\) Relationship: each Pattern B term is twice the corresponding Pattern A term.
5170594
Continue both number patterns logically, and find every sum. 1. \(210{,}000+320{,}000=\underline{\hspace{1cm}}\) 2. \(220{,}000+330{,}000=\underline{\hspace{1cm}}\) 3. \(230{,}000+340{,}000=\underline{\hspace{1cm}}\) 4. \(\underline{\hspace{1cm}}+\underline{\hspace{1cm}}=\underline{\hspace{1cm}}\) 5. \(\underline{\hspace{1cm}}+\underline{\hspace{1cm}}=\underline{\hspace{1cm}}\)

Hints

- Determine the change in each addend separately. - Apply both rules to create the next pair of addends. - When both addends increase by the same amount, consider how much the sum increases.

Solution

1. Each first addend increases by \(10{,}000\), and each second addend also increases by \(10{,}000\). 2. The first three sums are \(530{,}000\), \(550{,}000\), and \(570{,}000\). 3. Continue both addend patterns to get \(240{,}000\) and \(350{,}000\), whose sum is \(590{,}000\). 4. Continue once more to get \(250{,}000\) and \(360{,}000\), whose sum is \(610{,}000\).

Answer

1. \(530{,}000\) 2. \(550{,}000\) 3. \(570{,}000\) 4. \(240{,}000+350{,}000=590{,}000\) 5. \(250{,}000+360{,}000=610{,}000\)
5509925
Pattern A starts at \(10\) and follows the rule “add \(5\).” Pattern B starts at \(4\) and follows the rule “add \(8\).” The last row of the table is missing its position and Pattern A term. Complete that row. <table><tr><th>Position</th><th>Pattern A</th><th>Pattern B</th></tr><tr><td>1</td><td>\(10\)</td><td>\(4\)</td></tr><tr><td>2</td><td>\(15\)</td><td>\(12\)</td></tr><tr><td>3</td><td>\(20\)</td><td>\(20\)</td></tr><tr><td></td><td></td><td>\(28\)</td></tr></table>

Hints

- Work backward from the shown Pattern B term by locating it in Pattern B's repeated add rule. - Once you know the position, use the same position in Pattern A. - Check that both completed patterns follow their stated rules from one row to the next.

Solution

1. Pattern B goes \(4, 12, 20, 28\), so \(28\) is the fourth term. 2. Pattern A goes \(10, 15, 20, 25\), so its fourth term is \(25\). 3. The missing row is position \(4\) with corresponding terms \(25\) and \(28\).

Answer

Position: \(4\) Pattern A term: \(25\)
5509935
Pattern A starts at \(5\) and follows the rule “add \(4\).” Pattern B starts at \(8\) and follows the rule “add \(7\).” A student says, “The first terms differ by \(3\), so every pair of corresponding terms will differ by \(3\).” Is the student correct? Use the first four pairs of terms to explain.

Hints

- Generate several terms from both rules before deciding whether the claim is true. - Compare terms that occupy the same position in the two patterns. - Notice how the two add rules differ and consider what that does to the gap between the patterns.

Solution

1. The first four Pattern A terms are \(5, 9, 13, 17\). 2. The first four Pattern B terms are \(8, 15, 22, 29\). 3. The corresponding differences are \(3, 6, 9, 12\), so the difference does not stay \(3\). 4. Because Pattern B increases by \(3\) more than Pattern A at each step, the difference between corresponding terms increases by \(3\) each position.

Answer

The student is not correct. The first four corresponding differences are \(3, 6, 9, 12\), so the difference increases by \(3\) each position.
5509945
A growing tile design uses red and blue tiles. Design \(1\) has \(3\) red tiles and \(6\) blue tiles. For each new design, the number of red tiles increases by \(2\) and the number of blue tiles increases by \(4\). Complete the table for Designs \(1\) through \(4\). Then write an ordered pair \((\text{red}, \text{blue})\) for each design. <table><tr><th>Design</th><th>Red tiles</th><th>Blue tiles</th></tr><tr><td>1</td><td>\(3\)</td><td>\(6\)</td></tr><tr><td>2</td><td></td><td></td></tr><tr><td>3</td><td></td><td></td></tr><tr><td>4</td><td></td><td></td></tr></table>

Hints

- Extend the red and blue patterns separately using their two different add rules. - Match terms from the same design number when making an ordered pair. - Keep red first and blue second in every ordered pair.

Solution

1. Red-tile pattern: \(3, 5, 7, 9\). 2. Blue-tile pattern: \(6, 10, 14, 18\). 3. Pair corresponding terms in the order \((\text{red}, \text{blue})\): \((3, 6)\), \((5, 10)\), \((7, 14)\), and \((9, 18)\).

Answer

Completed red-tile pattern: \(3, 5, 7, 9\) Completed blue-tile pattern: \(6, 10, 14, 18\) Ordered pairs: \((3, 6)\), \((5, 10)\), \((7, 14)\), \((9, 18)\)
5203906
Lucas builds staircases from cubes. Each step is two cubes wide. He records his observations in a table: <table> <tr><td>Staircase height</td><td>\(1\)</td><td>\(2\)</td><td>\(3\)</td></tr> <tr><td>Cubes in the bottom row</td><td>\(2\)</td><td>\(4\)</td><td>\(6\)</td></tr> <tr><td>Total number of cubes</td><td>\(2\)</td><td>\(6\)</td><td>\(12\)</td></tr> </table> a) For a staircase with height \(10\), how many cubes are in the bottom row, and how many cubes are needed altogether? b) Let \(h\) be the staircase height, \(b\) the number of cubes in the bottom row, and \(t\) the total number of cubes. Write equations that express \(b\) and \(t\) in terms of \(h\).

Hints

- Compare the staircase height with the bottom-row count in each table column. - For the total, examine how the accumulated number of cubes changes as the height increases. - In part b), write each changing cube count as a dependent quantity determined by \(h\).

Solution

a) The bottom row has twice as many cubes as the height, so \(b = 2 \times 10 = 20\). The row counts are \(2, 4, 6, \ldots, 20\), whose total is \(110\) cubes. b) The bottom-row relationship is \(b = 2h\). The total number of cubes is the sum of the first \(h\) even numbers, which is \(t = h(h + 1)\). For \(h = 10\), this gives \(t = 10 \times 11 = 110\).

Answer

a) Bottom row: \(20\) cubes; total: \(110\) cubes b) \(b = 2h\) and \(t = h(h + 1)\)
5509955
Pattern A starts at \(6\) and follows the rule “add \(4\).” Pattern B starts at \(15\) and follows the rule “add \(7\).” At the first position, Pattern B is \(9\) greater than Pattern A. Without writing a variable formula, determine whether the difference between corresponding terms will ever be \(30\). If it will, state the position and explain why.

Hints

- Compare how much the two patterns increase from one position to the next. - Use the change in the gap between corresponding terms rather than generating both full patterns. - Count positions carefully: the starting pair is position \(1\).

Solution

1. Pattern B increases by \(7\) each step while Pattern A increases by \(4\), so the difference between corresponding terms increases by \(3\) each position. 2. Starting from a difference of \(9\), the differences are \(9, 12, 15, 18, 21, 24, 27, 30\). 3. The difference reaches \(30\) at position \(8\).

Answer

Yes. The difference is \(30\) at position \(8\).
5540665
Pattern A starts at \(4\) and follows the rule “add \(3\).” Pattern B starts at \(10\) and follows the rule “add \(6\).” Generate the first four terms of each pattern. Describe a relationship between corresponding terms, and explain why that relationship continues as both patterns grow.

Hints

- Generate corresponding terms in the same positions before comparing them. - Look for a relationship that combines multiplication and a fixed adjustment. - Compare the two add rules to explain why the relationship keeps working.

Solution

1. Pattern A is \(4, 7, 10, 13\), and Pattern B is \(10, 16, 22, 28\). 2. Each Pattern B term is \(2\) more than twice the corresponding Pattern A term. 3. The relationship starts correctly because \(10\) is \(2\) more than twice \(4\). Each time Pattern A increases by \(3\), twice its value increases by \(6\), which matches Pattern B’s “add \(6\)” rule. Therefore, the same relationship continues.

Answer

Pattern A: \(4, 7, 10, 13\) Pattern B: \(10, 16, 22, 28\) Relationship: each Pattern B term is \(2\) more than twice the corresponding Pattern A term. This continues because Pattern B increases by twice the amount that Pattern A increases.
5540675
Pattern A starts at \(3\) and follows the rule “add \(2\).” Pattern B starts at \(7\). The relationship should be: each Pattern B term is \(2\) less than \(3\) times the corresponding Pattern A term. What add rule must Pattern B follow so that this relationship continues? Generate the first four terms of both patterns to verify your rule, and explain why it works.

Hints

- Use the stated relationship to determine what the next few Pattern B terms would have to be. - Compare consecutive Pattern B terms to identify its add rule. - Explain how the multiplier in the relationship changes the size of Pattern A’s add rule.

Solution

1. Pattern A begins \(3, 5, 7, 9\). 2. If each Pattern B term is \(2\) less than \(3\) times the corresponding Pattern A term, the matching terms are \(7, 13, 19, 25\). 3. Pattern B therefore follows the rule “add \(6\).” 4. This works because each increase of \(2\) in Pattern A makes \(3\) times Pattern A increase by \(6\); subtracting the fixed \(2\) does not change that increase.

Answer

Pattern B must follow the rule “add \(6\).” The first four terms are: Pattern A: \(3, 5, 7, 9\) Pattern B: \(7, 13, 19, 25\) The relationship continues because tripling each \(2\)-unit increase in Pattern A gives a \(6\)-unit increase in Pattern B.

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