Discover Mathematics P5: Term 2, Unit 10, Lessons 9–11
GoalMake ordered pairs from two number patterns, plot them, and describe how the two patterns are related.
New words
pattern · numbers that follow a rule, such as add 2 each timerule · what you do to get from one term to the nextterm · one number in a patternrelationship · how the terms of two patterns are connected, such as “double”
Explanation
A number pattern follows a rule. Pattern 1: start at 0 and add 1: 0, 1, 2, 3, 4. Pattern 2: start at 0 and add 2: 0, 2, 4, 6, 8. Put the terms that are in the same step together and you get ordered pairs: (0, 0), (1, 2), (2, 4), (3, 6), (4, 8). The first number comes from Pattern 1 and the second from Pattern 2. Now plot these points. They all lie on a straight line that starts at the origin, because the rule between the two patterns never changes. Here every second number is double its first number, so y = 2 × x. When you see how the points line up, you can predict the next pair without calculating all the steps: (5, 10) comes next.
Step
Pattern 1 (+1)
Pattern 2 (+2)
Ordered pair
0
0
0
(0, 0)
1
1
2
(1, 2)
2
2
4
(2, 4)
3
3
6
(3, 6)
4
4
8
(4, 8)
Worked examples
Plot (1, 2), (2, 4), (3, 6) and (4, 8). The points make a straight line going up. Each time x grows by 1, y grows by 2.
Pattern 1 starts at 2 and adds 2: 2, 4, 6, 8. Pattern 2 starts at 1 and adds 2: 1, 3, 5, 7. The pairs are (2, 1), (4, 3), (6, 5), (8, 7). Each second number is 1 less than the first, so y = x − 1.
A baker uses 3 eggs for each cake. For 1, 2, 3 and 4 cakes the eggs are 3, 6, 9 and 12, so the pairs are (1, 3), (2, 6), (3, 9), (4, 12). For 7 cakes he needs 7 × 3 = 21 eggs: the pair (7, 21).
Class activity
Pattern partners. Use two dice or number cards. Choose a rule for Pattern 1 (add 1) and a rule for Pattern 2 (add 3). Write the first 5 terms of each, form the ordered pairs, and plot them on squared paper with a ruler. Do the points make a straight line? Change the rule of Pattern 2 to “add 4” and compare the two lines.
Practice
1
Pattern 1: start at 0, add 3. Pattern 2: start at 0, add 6. Write the first four ordered pairs.
(0, 0), (3, 6), (6, 12), (9, 18).
2
Pattern 1: start at 0, add 5. Pattern 2: start at 0, add 10. Write the next three pairs after (0, 0) and say how the numbers are related.
(5, 10), (10, 20), (15, 30). The second number is double the first.
3
The pairs are (1, 4), (2, 8), (3, 12). Find the rule and the next pair.
The second number is 4 times the first. Next pair: (4, 16).
4
The pairs are (2, 1), (4, 2), (6, 3), (8, 4). How are the two numbers related? What is the next pair?
The second number is half of the first. Next pair: (10, 5).
5
The graph shows three points of a pattern. Which point comes next if the pattern continues?
The points are (1, 3), (2, 6), (3, 9): y is 3 times x. Next: (4, 12).
