Two number patterns side by side and graphing their pairs
Two patterns grow side by side, each from its own rule. How are their matching terms related, and what does that look like when the pairs are drawn on a grid?
Where you are 4 in The coordinate plane
When you reach for it
A rule such as “start at 0, add 3” builds a number pattern one term at a time; that is Number and shape patterns from a rule. Often two things grow at the same time, each by its own rule. Takumi and Yifan each keep a caterpillar in a jar. Takumi’s caterpillar eats 3 leaves a day and Yifan’s eats 6 leaves a day. Each of them counts the total number of leaves eaten so far, starting from 0. That gives two patterns, one per caterpillar, growing day by day.
You reach for this skill when you want to know how two such patterns are related. Is one always bigger? By how much? Is one always a multiple of the other? Writing the two patterns side by side, pairing their matching terms, and plotting the pairs as points on a grid answers these questions, and the picture often makes the answer plain at a glance.
The lesson takes three days. Part 1 builds two patterns from two rules and pairs their matching terms. Part 2 finds the relationship between matching terms and explains why it holds at every step. Part 3 graphs the pairs as Ordered pairs and the first quadrant, reads the graph, and looks at patterns that do not start at 0 or whose relationship is not a plain multiple.
The procedure
Part 1: two rules, two patterns, matching terms
A rule names a starting number and what to do to get from each term to the next. The two caterpillar rules are:
- Takumi’s pattern: start at 0, add 3.
- Yifan’s pattern: start at 0, add 6.
Each rule is used over and over, so each pattern can go on as long as you like. Write them in one table, with the number of days so far on top, so that terms made on the same day sit in the same column.
| Days so far | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| Takumi’s total | 0 | 3 | 6 | 9 | 12 | 15 |
| Yifan’s total | 0 | 6 | 12 | 18 | 24 | 30 |
Two terms in the same column are matching terms, or corresponding terms: they are in the same position in their patterns. The 1st term of one goes with the 1st term of the other, the 2nd with the 2nd, and so on. Here, after 4 days, Takumi’s total is 12 and Yifan’s is 24, so 12 and 24 are matching terms.
Matching goes by position, not by value. Both patterns contain the number 12, but Takumi’s 12 is his 5th term (after 4 days) and Yifan’s 12 is her 3rd term (after 2 days). They are not matching terms. Takumi’s 12 is matched with Yifan’s 24.
Each pair of matching terms can be written as an ordered pair, with the first pattern’s term first:
The order is a choice, and it must be stated: here the first number is always Takumi’s total and the second is always Yifan’s.
Part 2: the relationship between matching terms
Look down each column of the table. In every column, Yifan’s total is twice Takumi’s:
Read the other way, Takumi’s total is always half of Yifan’s. This is a relationship between corresponding terms, one from each pattern. It is a different thing from the rule of either pattern. The rules say how each pattern grows along its own row; the relationship says how the two rows compare in each column.
Checking six columns shows the relationship in those columns. It does not yet show that it holds on day 100. The reason it always holds comes from the rules themselves.
After some number of days, say 4, Takumi’s caterpillar has eaten 3 leaves on each of 4 days:
Yifan’s has eaten 6 leaves on each of the same 4 days, and each 6 is 2 \times 3:
Multiplication can be reordered (the commutative property) and regrouped (the associative property) without changing the product, so
Nothing in that argument used the number 4 in a special way. After any number of days, Yifan’s total is that many sixes, and each six is two threes, so her total is twice Takumi’s. Put another way: both start at 0, which is twice 0, and each day Yifan’s total grows by 6 while Takumi’s grows by 3. Adding 6 to one side and 3 to the other keeps “twice” true, because if Yifan has twice as many, then after adding 6 she still has twice as many as Takumi after he adds 3.
The same reasoning works for any two rules that both start at 0 and add a fixed number: each term of the second pattern is the same multiple of the matching term of the first, and that multiple is the second step divided by the first step. For add 3 and add 6 it is 6 \div 3 = 2.
Part 3: graphing the pairs
Each pair of matching terms is an ordered pair, so it names a point on the grid. Plot the first number across and the second number up, as in Ordered pairs and the first quadrant. Because the numbers grow quickly, both axes here count by 3s.
Three things show up in the picture.
The points line up. From one point to the next, Takumi’s total goes up by 3 and Yifan’s by 6, so every move on the grid is the same: 3 across, then 6 up. The dashed staircase shows these moves. Equal steps in the same direction keep the points on one straight line.
The line starts at the origin. Both patterns start at 0, so the first pair is (0, 0).
The line is steep. Each point is twice as far up as it is across, because Yifan’s total is twice Takumi’s. For every 1 unit across, the points rise 2 units.
The graph can be read backward too. The point (9, 18) says: on the day Takumi’s total was 9, Yifan’s was 18. Following the line one more step from (15, 30) gives (18, 36), the totals after 6 days.
Patterns that start at other numbers, or whose steps are not a simple multiple of each other, also give points in a line when both rules add a fixed number, but the line may not pass through the origin, and the relationship between matching terms can be “always 6 more” or “twice, then add 1” rather than a plain multiple. Examples 6 and 7 work through both.
Worked examples
Examples 1 and 2 go with Part 1, Examples 3 and 4 with Part 2, and Examples 5 to 7 with Part 3.
Rides and tickets
At the amusement park, the bumper cars cost 4 tickets a ride. Gwyneth keeps two patterns: the number of rides she has taken (start at 0, add 1) and the number of tickets she has used (start at 0, add 4). Write the first six terms of each, and the ordered pairs of matching terms.
SolutionEach pattern starts at its starting number, and the rule is used again for each new term.
- Rides: 0, 1, 2, 3, 4, 5, adding 1 each time.
- Tickets: 0, 4, 8, 12, 16, 20, adding 4 each time.
Matching terms are the ones in the same position. The 4th term of the rides pattern is 3 and the 4th term of the tickets pattern is 12: after 3 rides she has used 12 tickets. Writing rides first and tickets second, the pairs are
Here the first pattern simply counts the rides, so each pair reads “after this many rides, this many tickets”.
Finding the matching term
Yifan’s rule is “start at 0, add 2” and Takumi’s is “start at 0, add 8”. Which term of Yifan’s pattern is 14? What is the matching term of Takumi’s pattern? Write the pair.
SolutionTo find the position of 14, count how many times 2 was added to get from 0 to 14: 14 \div 2 = 7. Seven additions after the starting term means 14 is the 8th term. Listing confirms it: 0, 2, 4, 6, 8, 10, 12, 14 has eight terms.
The matching term of Takumi’s pattern is also the 8th term, so it too comes after seven additions, this time of 8:
The pair is (14, 56). Takumi’s pattern begins 0, 8, 16, \ldots, and none of its early terms is the match: only the term in the same position, after seven additions, is.
Finding the relationship and saying why
The rules are “start at 0, add 4” and “start at 0, add 12”. Describe the relationship between corresponding terms, and explain why it holds for every term.
SolutionList several terms of each.
- First pattern: 0, 4, 8, 12, 16, 20.
- Second pattern: 0, 12, 24, 36, 48, 60.
Compare each column: 12 = 3 \times 4, 24 = 3 \times 8, 36 = 3 \times 12, and so on. Each term of the second pattern is 3 times the matching term of the first.
Why it always holds: after any number of steps, the first pattern is that many fours and the second is that many twelves. Each twelve is three fours, since 12 = 3 \times 4, so that many twelves is three times that many fours. After 5 steps, for example,
using the commutative and associative properties of multiplication. The multiple is the ratio of the steps, 12 \div 4 = 3. Read the other way, each term of the first pattern is one third of the matching term of the second.
A relationship that is not a whole number
Yifan’s caterpillar eats 4 leaves a day and Takumi’s new caterpillar eats 10 leaves a day; both totals start at 0. Describe the relationship between their totals on the same day. When Yifan’s caterpillar has eaten 28 leaves, how many has Takumi’s eaten?
SolutionThe patterns are 0, 4, 8, 12, 16 and 0, 10, 20, 30, 40. Compare a column: 10 is 2 fours and half a four more, since 2 \times 4 + 2 = 10. So Takumi’s total is \displaystyle 2\frac{1}{2} times Yifan’s. The other columns agree: \displaystyle 2\frac{1}{2} \times 8 = 16 + 4 = 20 and \displaystyle 2\frac{1}{2} \times 12 = 24 + 6 = 30. The reason is the same as before: each day adds 10 to one total and 4 to the other, and \displaystyle 10 = 2\frac{1}{2} \times 4.
Now find Takumi’s total when Yifan’s is 28. Yifan’s caterpillar eats 4 a day, so 28 leaves took 28 \div 4 = 7 days. In the same 7 days Takumi’s ate
Check with the relationship: \displaystyle 2\frac{1}{2} \times 28 = 56 + 14 = 70. Both ways give 70.
Graphing rides and tickets
Graph Gwyneth’s pairs from Example 1, rides across and tickets up. Describe the points. Then use the pattern of the points to say how many tickets 7 rides would use.
SolutionThe pairs are (0, 0), (1, 4), (2, 8), (3, 12), (4, 16), (5, 20). The tickets reach 20, so the up axis counts by 2s and fits on 10 squares; the across axis counts by 1s.
Each move from one point to the next is 1 ride across and 4 tickets up, which is 2 squares up on this axis. The move is the same every time, so the points lie on one straight line, and since both patterns start at 0 the line starts at the origin. The tickets are always 4 times the rides.
To reach 7 rides, take two more moves past (5, 20): (6, 24), then (7, 28). So 7 rides use 28 tickets, which agrees with 4 \times 7 = 28.
Patterns that do not start at 0
At the carousel each ride costs 2 tickets. Gwyneth had already used 6 tickets on other rides when she and Yifan reached the carousel, and Yifan had used none. They ride the carousel together, ride after ride. Write the two patterns of tickets used, pair them with Yifan’s first, graph the pairs, and describe the relationship.
SolutionYifan’s rule is “start at 0, add 2”: 0, 2, 4, 6, 8. Gwyneth’s is “start at 6, add 2”: 6, 8, 10, 12, 14. The pairs are
Compare each column: 6 = 0 + 6, 8 = 2 + 6, 10 = 4 + 6. Gwyneth’s total is always 6 more than Yifan’s. It cannot be a multiple: in the first column Yifan’s term is 0, and every multiple of 0 is 0, not 6. The reason for “6 more” is that Gwyneth started 6 ahead and both add the same 2 each ride, so the gap never changes.
On the graph the points still line up, because each move is the same: 2 across and 2 up. But the first point is (0, 6), on the y-axis, so the line does not pass through the origin. Every point is 6 units higher than it would be if the two totals were equal.
Twice, then add one
The first pattern is “start at 0, add 2” and the second is “start at 1, add 4”. Describe the relationship between corresponding terms, explain why it holds, and say what the graph of the pairs looks like.
SolutionThe patterns are 0, 2, 4, 6, 8 and 1, 5, 9, 13, 17, giving the pairs
The second term is not a fixed multiple of the first, since 1 is not a multiple of 0. Nor is it a fixed amount more: the gaps 1, 3, 5, 7, 9 keep growing. Try doubling the first term and comparing: 2 \times 0 = 0, 2 \times 2 = 4, 2 \times 4 = 8, 2 \times 6 = 12. Each second term is exactly 1 more than that: 1, 5, 9, 13. So each term of the second pattern is twice the matching term of the first, plus 1.
Why it holds at every step: it is true at the start, since 1 = 2 \times 0 + 1. At each step the first term grows by 2, so twice the first term grows by 4; and the second term grows by exactly 4. Both sides of “second = twice first + 1” grow by the same 4, so the statement stays true.
Graphed with the first pattern across, each move is 2 across and 4 up, so the points line up, rising 2 units for every 1 across like the caterpillar graph. But the line starts at (0, 1), one unit above the origin.
Boundaries
The two patterns must have the same number of terms, or some terms have no partner. Matching is by position: the 5th term goes with the 5th term, even when some other term of the second pattern happens to equal a term of the first.
A relationship seen in a few columns is a guess until the rules explain it. The rules “start at 0, add 2” and “start at 0, add 5” give (0, 0) and (2, 5), and two columns alone fit many relationships; it is the steps, \displaystyle 5 = 2\frac{1}{2} \times 2, that fix the relationship for every term.
A plain multiple needs both patterns to start at 0, or more generally needs the second start to be that same multiple of the first start. When the starts do not fit, the relationship may be “a fixed amount more”, as in Example 6, or a mix, as in Example 7.
The points in this lesson line up because both rules add a fixed number, so every move is the same. When one rule adds and the other multiplies, the moves are usually not the same. Pair “start at 0, add 1” with “start at 1, multiply by 2”: the pairs are (0, 1), (1, 2), (2, 4), (3, 8), and the moves up are 1, 2, 4, so the points bend upward instead of lining up.
The graph of two patterns is the set of points, one for each pair. A line drawn through them is a guide for the eye. A point on that line between two plotted points, such as \displaystyle (4\frac{1}{2}, 9) on the caterpillar line, is not one of the pairs, because \displaystyle 4\frac{1}{2} is not a term of “start at 0, add 3”.
Every number here is 0 or more, so every point is in the first quadrant.
Where this goes next
Pairing two patterns is the first step toward describing how one quantity depends on another. In Graphing a line from a table of values, a table of pairs from a real situation is graphed in the same way, and the straight line through the origin seen here returns as the picture of two quantities in a fixed ratio. Plotting in all four quadrants lets the grid run past 0, so patterns that count down below zero can be graphed too. All of these pictures live on the The coordinate plane.
Exercises
Part 1: two patterns and matching terms
- Write the first six terms of “start at 0, add 4” and of “start at 0, add 8”. Write the ordered pairs of matching terms.
- Write the first five terms of “start at 0, add 5” and of “start at 0, add 2”, and the ordered pairs of matching terms, with the first rule’s term first.
- Takumi’s caterpillar eats 4 leaves a day and Yifan’s eats 7. Both totals start at 0. Write each total for 0 to 4 days so far, and the pairs (Takumi’s total, Yifan’s total).
- Gwyneth starts a table. The first row is “start at 0, add 6”: 0, 6, 12, \_, \_. The second row is “start at 0, add 9”: 0, 9, \_, 27, \_. Fill in the four blanks.
- The rules are “start at 0, add 3” and “start at 0, add 10”. Which term of the first pattern is 21? Write the pair it belongs to.
- Yifan wrote the pairs (0, 0), (2, 5), (4, 10), (6, 15) from two rules. What were the rules?
- Is (12, 30) one of the pairs made from “start at 0, add 4” and “start at 0, add 10”?
- The rules are “start at 0, add 3” and “start at 0, add 9”. Takumi pairs the 9 in the first pattern with the 9 in the second, and writes (9, 9). What is the correct pair for the first pattern’s 9?
Part 2: the relationship between matching terms
- The rules are “start at 0, add 5” and “start at 0, add 20”. Describe the relationship between corresponding terms.
- The rules are “start at 0, add 15” and “start at 0, add 5”. Describe the relationship between corresponding terms both ways.
- The rules are “start at 0, add 6” and “start at 0, add 9”. Describe the relationship between corresponding terms.
- The rules are “start at 0, add 8” and “start at 0, add 16”. What term of the second pattern matches the 56 in the first?
- The rules are “start at 0, add 3” and “start at 0, add 15”. What term of the first pattern matches the 90 in the second?
- For the rules “start at 0, add 2” and “start at 0, add 6”, Yifan says the second term is always 4 more than the first, because 6 - 2 = 4. Is she right? Give the correct relationship.
- The first rule is “start at 0, add 4”. Write a second rule so that each term of the second pattern is 5 times the matching term of the first.
Part 3: graphs and other starts
Exercises 16 to 18 use this graph of two patterns, the first pattern across and the second up.
- List the ordered pairs shown on the graph.
- What rules made the two patterns? Describe the relationship between corresponding terms.
- Following the pattern of the points, what are the next two points?
- Pair “start at 0, add 1” with “start at 0, add 3” for five terms and graph the pairs. Describe the move from one point to the next and where the line of points starts.
- Pair “start at 0, add 3” with “start at 5, add 3” for four terms. Describe the relationship and where the line of points meets the y-axis.
- Pair “start at 0, add 1” with “start at 2, add 3” for four terms. Describe the relationship between corresponding terms, and compare the graph with the one in Exercise 19.
Reasoning
- Two rules both start at 0, and at each step the second adds 4 times as much as the first. Explain why every term of the second pattern is 4 times the matching term of the first, whatever number the first rule adds.
- Pair “start at 0, add 1” with “start at 1, multiply by 2” for four terms. Explain why the points do not line up.
- For the caterpillars, Takumi graphs the pairs as (Yifan’s total, Takumi’s total) instead. What changes in the graph, and what relationship does it show?
- For the rules “start at 0, add 3” and “start at 0, add 6”, every second term is twice the first, and 14 = 2 \times 7. Explain why (7, 14) is still not a point on the graph, and why no term of the second pattern is odd.
Answers
- First: 0, 4, 8, 12, 16, 20. Second: 0, 8, 16, 24, 32, 40. Pairs (0, 0), (4, 8), (8, 16), (12, 24), (16, 32), (20, 40).
- First: 0, 5, 10, 15, 20. Second: 0, 2, 4, 6, 8. Pairs (0, 0), (5, 2), (10, 4), (15, 6), (20, 8).
- Takumi: 0, 4, 8, 12, 16. Yifan: 0, 7, 14, 21, 28. Pairs (0, 0), (4, 7), (8, 14), (12, 21), (16, 28).
- First row 0, 6, 12, 18, 24; second row 0, 9, 18, 27, 36. The blanks are 18, 24, 18 and 36.
- The 8th term, since 21 \div 3 = 7 additions. The matching term is 7 \times 10 = 70, so the pair is (21, 70).
- First: start at 0, add 2. Second: start at 0, add 5. Each pattern starts at 0, and its terms go up by the same amount each time.
- Yes. 12 \div 4 = 3 additions, and 3 \times 10 = 30, so 30 is the matching term.
- (9, 27). The first pattern’s 9 comes after 9 \div 3 = 3 additions, and the second pattern’s term after 3 additions is 3 \times 9 = 27. The second pattern’s 9 is its 2nd term, a different position.
- Each term of the second pattern is 4 times the matching term of the first, since 20 = 4 \times 5: 0, 5, 10, 15 and 0, 20, 40, 60.
- Each term of the first is 3 times the matching term of the second, since 15 = 3 \times 5; each term of the second is one third of the matching term of the first. The patterns are 0, 15, 30, 45 and 0, 5, 10, 15.
- Each term of the second is \displaystyle 1\frac{1}{2} times the matching term of the first, since \displaystyle 9 = 1\frac{1}{2} \times 6. Check: 0, 6, 12, 18 against 0, 9, 18, 27.
- 112. 56 \div 8 = 7 additions, and 7 \times 16 = 112, which is 2 \times 56.
- 18. 90 \div 15 = 6 additions, and 6 \times 3 = 18; check 5 \times 18 = 90.
- No. The patterns 0, 2, 4, 6 and 0, 6, 12, 18 differ by 0, 4, 8, 12, a gap that grows. Each term of the second is 3 times the first, since 6 = 3 \times 2.
- Start at 0, add 20, since 5 \times 4 = 20 and both patterns start at 0.
- (0, 0), (2, 3), (4, 6), (6, 9), (8, 12).
- First: start at 0, add 2. Second: start at 0, add 3. Each second term is \displaystyle 1\frac{1}{2} times the first, since \displaystyle 3 = 1\frac{1}{2} \times 2.
- (10, 15) and (12, 18), each one move of 2 across and 3 up past the last.
- Pairs (0, 0), (1, 3), (2, 6), (3, 9), (4, 12). Each move is 1 across and 3 up, so the points line up, starting at the origin. Each second term is 3 times the first.
- Pairs (0, 5), (3, 8), (6, 11), (9, 14). Each second term is 5 more than the first. The points line up and meet the y-axis at (0, 5).
- Pairs (0, 2), (1, 5), (2, 8), (3, 11). Each second term is 3 times the first, plus 2. The moves are 1 across and 3 up, as in Exercise 19, so the line is just as steep, but it starts at (0, 2), 2 units above the origin.
- Call the first rule’s step the first number. After some number of additions, the first pattern is that many copies of the first number. The second pattern is the same count of copies of its own step, and each of those is 4 copies of the first number. So the second pattern holds 4 times as many copies, which is 4 times the first term; regrouping like this is the associative and commutative properties of multiplication. The starting terms are 0 and 0, and 0 = 4 \times 0, so the first column fits too.
- The pairs are (0, 1), (1, 2), (2, 4), (3, 8). Each move is 1 across, but the moves up are 1, 2, 4, which grow. Points line up only when every move is the same, and a multiplying rule makes the steps grow.
- The two numbers in every pair swap, giving (0, 0), (6, 3), (12, 6), and so on. The points still line up through the origin, but now each point is half as far up as it is across, so the line is less steep. It shows that each of Takumi’s totals is half of Yifan’s, the same relationship read the other way.
- A point needs a first number that is a term of the first pattern. The terms of “start at 0, add 3” are 0, 3, 6, 9, \ldots, multiples of 3, and 7 is not one, so no pair starts with 7. Every term of the second pattern is 6 times a whole number, and 6 times anything whole is even, since 6 = 2 \times 3; so no term is odd.