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Graph ordered pairs from patterns

Learning goals

  • Forms ordered pairs from corresponding terms of two patterns.
  • Plots ordered pairs from corresponding terms of the patterns.
  • Describes a relationship between two patterns from the plotted points.

Click problems to add them to your worksheet.

Graph ordered pairs from patterns 1
The table shows corresponding terms from two number patterns. Use Pattern A as the x-coordinate and Pattern B as the y-coordinate. Write the four ordered pairs. <table><tr><th>Position</th><th>Pattern A</th><th>Pattern B</th></tr><tr><td>1</td><td>\(1\)</td><td>\(2\)</td></tr><tr><td>2</td><td>\(3\)</td><td>\(5\)</td></tr><tr><td>3</td><td>\(5\)</td><td>\(8\)</td></tr><tr><td>4</td><td>\(7\)</td><td>\(11\)</td></tr></table>

Hints

- An ordered pair is written as \((x, y)\). - Use the Pattern A entry in a row as the first coordinate. - Use the Pattern B entry from the same row as the second coordinate.

Solution

1. Pair the terms in each row with Pattern A first and Pattern B second. 2. The ordered pairs are \((1, 2)\), \((3, 5)\), \((5, 8)\), and \((7, 11)\).

Answer

\((1, 2)\), \((3, 5)\), \((5, 8)\), \((7, 11)\)
Graph ordered pairs from patterns 2
The corresponding terms from Pattern A and Pattern B are already written as ordered pairs \((\text{Pattern A}, \text{Pattern B})\): \((1, 5)\), \((3, 2)\), and \((5, 4)\). Plot all three points on the coordinate plane. Which plotted point is highest?
Figure for problem 558946

Hints

- The first coordinate gives the horizontal position and the second gives the vertical position. - Plot one ordered pair at a time before comparing the points. - For “highest,” compare the vertical positions of the plotted points.

Solution

1. Plot \((1, 5)\), \((3, 2)\), and \((5, 4)\) using the first coordinate for the x-axis and the second for the y-axis. 2. The point \((1, 5)\) has the greatest y-coordinate, so it is highest.

Answer

\((1, 5)\)
Graph ordered pairs from patterns 3
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: \((3, 6)\), \((5, 10)\), \((7, 14)\), \((9, 18)\)
Graph ordered pairs from patterns 4
The table lists four ordered pairs made from corresponding terms of Pattern A and Pattern B. Pattern A is the x-coordinate and Pattern B is the y-coordinate. The graph shows only three of the four points. Which ordered pair from the table is missing from the graph? State where it should be plotted. <table><tr><th>Position</th><th>Pattern A</th><th>Pattern B</th></tr><tr><td>1</td><td>\(4\)</td><td>\(1\)</td></tr><tr><td>2</td><td>\(7\)</td><td>\(3\)</td></tr><tr><td>3</td><td>\(10\)</td><td>\(5\)</td></tr><tr><td>4</td><td>\(13\)</td><td>\(7\)</td></tr></table>
Figure for problem 550999

Hints

- Turn each table row into an ordered pair with Pattern A first. - Compare the four table pairs with the three points already shown on the graph. - The pair that does not appear on the graph is the one that must be plotted.

Solution

1. The table gives the ordered pairs \((4, 1)\), \((7, 3)\), \((10, 5)\), and \((13, 7)\). 2. The graph already shows \((4, 1)\), \((7, 3)\), and \((10, 5)\). 3. The missing point is \((13, 7)\), so it should be plotted at x-coordinate \(13\) and y-coordinate \(7\).

Answer

The missing ordered pair is \((13, 7)\). Plot it at x-coordinate \(13\) and y-coordinate \(7\).
Graph ordered pairs from patterns 5
Two patterns are shown. First pattern: \(2, 4, 6, \dots\) Second pattern: \(5, 8, 11, \dots\) Continue each pattern by one term. Use the two new terms to make the ordered pair \((\text{first pattern}, \text{second pattern})\), then plot that point on the coordinate plane.
Figure for problem 558947

Hints

- Find the change between neighboring terms in each pattern. - Continue both patterns to the same next position. - Use the first-pattern term as the x-coordinate and the second-pattern term as the y-coordinate.

Solution

1. The next first-pattern term is \(8\), and the next second-pattern term is \(14\). 2. The new corresponding ordered pair is \((8, 14)\). 3. Plot the point at x-coordinate \(8\) and y-coordinate \(14\).

Answer

\((8, 14)\)
Graph ordered pairs from patterns 6
These ordered pairs \((\text{Pattern A}, \text{Pattern B})\) came from corresponding terms of two patterns: \((2, 5)\), \((4, 10)\), \((6, 15)\), and \((8, 20)\). Plot all four points on the coordinate plane. Which ordered pair lands three grid intervals to the right and three grid intervals up from the origin?
Figure for problem 558949

Hints

- Read the value of one horizontal grid interval and one vertical grid interval. - Use the two axis scales separately. - Check which plotted ordered pair is at the resulting grid location.

Solution

1. On the x-axis, one grid interval represents \(2\). Three intervals represent \(6\). 2. On the y-axis, one grid interval represents \(5\). Three intervals represent \(15\). 3. The point three intervals right and three intervals up is \((6, 15)\).

Answer

\((6, 15)\)
Graph ordered pairs from patterns 7
Pattern A starts at \(2\) and follows the rule “add \(3\).” Pattern B starts at \(7\) and follows the rule “add \(2\).” Zofia plotted the first three corresponding ordered pairs, using Pattern A as x and Pattern B as y, but one point is incorrect. Identify the incorrect plotted point and give the coordinates it should have.
Figure for problem 551000

Hints

- Generate the first three terms of each pattern from the stated rules. - Pair terms from the same position with Pattern A first and Pattern B second. - Compare those expected ordered pairs with the three plotted points.

Solution

1. The first three Pattern A terms are \(2, 5, 8\), and the first three Pattern B terms are \(7, 9, 11\). 2. The correct ordered pairs are \((2, 7)\), \((5, 9)\), and \((8, 11)\). 3. The graph shows \((9, 5)\) instead of \((5, 9)\), so the coordinates of the second point were reversed.

Answer

The incorrect point is \((9, 5)\). It should be \((5, 9)\).
Graph ordered pairs from patterns 8
Pattern A starts at \(1\) and follows the rule “add \(3\).” Pattern B starts at \(4\) and follows the rule “add \(2\).” Find the ordered pairs \((\text{Pattern A}, \text{Pattern B})\) for positions \(2\), \(3\), and \(4\). Which coordinate plane correctly shows those three points?
Figure for problem 551001

Hints

- Generate the terms at positions \(2\), \(3\), and \(4\) in each pattern. - Keep Pattern A first when you form each ordered pair. - Compare all three ordered pairs with both coordinate planes before choosing.

Solution

1. Pattern A is \(1, 4, 7, 10\), and Pattern B is \(4, 6, 8, 10\). 2. The position-\(2\), position-\(3\), and position-\(4\) pairs are \((4, 6)\), \((7, 8)\), and \((10, 10)\). 3. Panel a) plots those three ordered pairs. Panel b) places the position-\(3\) point at \((7, 9)\) instead of \((7, 8)\).

Answer

Position \(2\): \((4, 6)\) Position \(3\): \((7, 8)\) Position \(4\): \((10, 10)\) Correct graph: panel a)
Graph ordered pairs from patterns 9
Pattern A starts at \(3\) and follows the rule “add \(4\).” Pattern B starts at \(5\) and follows the rule “add \(6\).” Renzo says the ordered pair \((23, 34)\) belongs to the graph of corresponding terms, with Pattern A as x and Pattern B as y. Decide whether Renzo is correct. If not, give the ordered pair that has x-coordinate \(23\) and justify your answer.

Hints

- Locate \(23\) in Pattern A by following its add rule. - Use the same position number in Pattern B; corresponding coordinates must come from the same position. - Check both coordinates against their pattern rules before accepting the proposed point.

Solution

1. Pattern A reaches \(23\) at the sixth position: \(3, 7, 11, 15, 19, 23\). 2. The first six Pattern B terms are \(5, 11, 17, 23, 29, 35\). 3. At the same sixth position, the corresponding Pattern B term is \(35\), not \(34\). 4. Therefore, \((23, 34)\) does not belong; the correct ordered pair with x-coordinate \(23\) is \((23, 35)\).

Answer

Renzo is not correct. The correct ordered pair is \((23, 35)\).
Graph ordered pairs from patterns 10
From one position to the next, Pattern A increases by \(2\) and Pattern B increases by \(3\). Use Pattern A as the x-coordinate and Pattern B as the y-coordinate. The graph shows the current plotted pair and four labeled candidate locations for the next pair. Which labeled location is correct? Give both the label and the ordered pair.
Figure for problem 554068

Hints

- Read the current ordered pair from the solid plotted point. - Apply the two stated increases to its x- and y-coordinates. - Use the axis tick labels to match the new ordered pair to one of the open labeled candidate points.

Solution

1. Read the current point from the graph: \((4, 7)\). 2. The next Pattern A term is \(4+2=6\), and the next Pattern B term is \(7+3=10\). 3. The next ordered pair is therefore \((6, 10)\). 4. On the graph, the candidate located at \((6, 10)\) is labeled C.

Answer

C; \((6, 10)\)
Graph ordered pairs from patterns 11
The graph shows the first four ordered pairs made from corresponding terms of Pattern A and Pattern B. Pattern A is on the x-axis and Pattern B is on the y-axis. Use the plotted points to state the starting number and add rule for each pattern. Then give the next ordered pair.
Figure for problem 554069

Hints

- Read the x-coordinates as one sequence and the y-coordinates as a second sequence. - Compare consecutive values on each axis to identify each add rule. - Extend both patterns one more position before forming the next ordered pair.

Solution

1. Reading the x-coordinates from left to right gives Pattern A: \(1, 3, 5, 7\). It starts at \(1\) and follows the rule “add \(2\).” 2. Reading the y-coordinates gives Pattern B: \(4, 7, 10, 13\). It starts at \(4\) and follows the rule “add \(3\).” 3. The next terms are \(9\) and \(16\), so the next ordered pair is \((9, 16)\).

Answer

Pattern A: start at \(1\), add \(2\). Pattern B: start at \(4\), add \(3\). Next ordered pair: \((9, 16)\)
Graph ordered pairs from patterns 12
Use these rules for positions \(1\) through \(4\). First pattern: start at \(1\), then add \(2\) each time. Second pattern: start at \(2\), then add \(3\) each time. Write the four ordered pairs \((\text{first pattern}, \text{second pattern})\) and plot all four points. Which plotted pair is highest?
Figure for problem 558948

Hints

- Generate the two patterns for the same four positions. - Form each ordered pair from terms in the same position. - After plotting all four pairs, compare their vertical positions.

Solution

1. The first pattern is \(1, 3, 5, 7\), and the second pattern is \(2, 5, 8, 11\). 2. The ordered pairs are \((1, 2)\), \((3, 5)\), \((5, 8)\), and \((7, 11)\). 3. Plot those four points. The pair \((7, 11)\) has the greatest y-coordinate, so it is highest.

Answer

\((1, 2)\), \((3, 5)\), \((5, 8)\), \((7, 11)\) Highest point: \((7, 11)\)

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