Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule "Add 3" and the starting number 0, and given the rule "Add 6" and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Here students run two number patterns side by side and look for a connection between them. Starting both at 0, the rule "add 3" gives 0, 3, 6, 9, 12 and the rule "add 6" gives 0, 6, 12, 18, 24. Lining up the terms in position shows that each term of the second pattern is twice the matching term of the first.
The real goal is to explain why the relationship holds. After the same number of steps, the second pattern has added 6 each time while the first has added 3, so it has always added exactly double. Students then pair corresponding terms as ordered pairs, (3, 6), (6, 12), (9, 18), and plot them on the coordinate plane, where the points line up in a straight line through the origin. This early look at two quantities changing together is the seed of ratio tables in sixth grade and linear functions in eighth grade.
Students sometimes compare the 3rd term of one list with the 4th term of the other. A two-row table with a position row makes corresponding terms easy to match.
Saying 'one goes up by 3 and the other by 6' describes each rule but not the relationship. Push for a statement linking matching terms, such as 'the second is always double the first'.
If the two patterns start at different numbers, a simple multiple may not hold. Students must test the relationship on every pair rather than assume it.
When plotting, the term from the first pattern is the x-coordinate. Reversing the order puts points in the wrong places and hides the straight line.
Start both patterns at 0. Pattern A uses the rule add 4 and pattern B uses add 12. List five terms of each, describe the relationship and give the ordered pair for the fifth terms.
Answer: Each term of pattern B is 3 times the corresponding term of pattern A; the fifth ordered pair is (16, 48).
Have students record both patterns in a two-row table and draw arrows between matching terms before they look for a relationship. Plotting the ordered pairs on grid paper and seeing them fall on a straight line gives a strong visual confirmation of the multiplicative link.
Questions on this standard typically give two rules and ask for a missing term, the relationship between corresponding terms, or which graph shows the ordered pairs.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: 10
The terms are 0, 2, 4, 6, 8, 10. After five steps of adding 2 the 6th term is 10.
Answer: B) B is 2 times A
10 = 2 × 5, 20 = 2 × 10 and 30 = 2 × 15. The differences 5, 10 and 15 are not constant, so 'more than' does not work.
Answer: 45
B adds 9 while A adds 3, so every B term is 3 times its A term. 3 × 15 = 45.
Answer: D) (3, 12)
The 4th terms are 3 and 12, giving (3, 12). The first number comes from pattern A.
Answer: 14
A adds 3 times as much as B each step, so each B term is one third of its A term: 42 ÷ 3 = 14.
Answer: B) Because after the same number of steps, adding 8 each time gives double the total of adding 4 each time
After n steps one pattern has added 8 n times and the other 4 n times. Since 8 = 2 × 4, the first total is always twice the second.
A full lesson with slides, activities and an exit ticket on patterns from two rules, pitched to grade 5 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 5.OA.B.3 with an answer key, ready in about a minute.
Make a worksheet →Turn patterns from two rules into a quiz students answer online that marks itself, with a class summary for you.
Build a test →The standard's example starts both at 0, which makes a multiplicative relationship appear. Patterns can start elsewhere, but then students should test whether a simple relationship really holds for every pair.
Corresponding terms become ordered pairs that are plotted in the first quadrant, linking this standard to the coordinate plane work in 5.G.A.1 and 5.G.A.2.