Numerical Patterns
How does making tables help you identify relationships between numbers in patterns ?
You can line up the numbers and compare them easily.
During summer vacation, Julie read 45 pages each day. Her brother Bret read 15 pages each day. Complete a table to show how many pages they each read after 8 days. What do you notice in the pattern?
After 8 days:
- Bret has read 120 pages
- Julie has read 360 Pages
Pattern: Julie reads 3 times as many pages as Bret, so she will always have 3 times the total number Bret has.
Each day Hank prepared 2 chicken pot pies and 10 blueberry pies at the soup kitchen. What ordered pair would represent the TOTAL number of each type of pie Hank makes in 7 days?
(Hint: X is the number of pot pies and Y is the number of blueberry pies)
(14,70)
On Aiden's farm, there are 12 acres of soybeans and 8 acres of corn. Aident plans to replace his other crops with more acres of soybeans and corn. Will his farm ever have the same number of acres of soybeans and corn? Explain.
His crops will be the same number of acres after 4 years. They will both have 24 acres of each crop after 4 years.
A dogwood tree exhibits opposite phyllotaxis (the way leaves face ) while a redbud tree shows alternating phyllotaxis. Greg has two tree seedlings. The dogwood seedling has 6 leaves, and the redbud seedling has 16 leaves. Both seedlings grow 4 new leaves each week. How many leaves did each seedling have after 6 weeks?
Dogwood: 30 leaves
Redbud: 40 leaves
Mike and Sarah are packing boxes at a factory. Mike packs 30 boxes each hour. Sarah packs 15 boxes each hour. How many boxes will each person have packed after an 8-hour shift?
Mike: 240 boxes
Sarah: 120 boxes
Maya is helping her school decorate for a festival. She is making rows of paper lanterns. In the first row, she hangs 3 lanterns. In the second row, she hangs 6 lanterns. In the third row, she hangs 9 lanterns, and she continues the pattern the same way for each new row.
First five rows:
3, 6, 9, 12, 15
Pattern:
The number of lanterns increases by 3 each time. Each row has 3 more lanterns than the row before it.
A school cafeteria is tracking how many lunches are served each day during a special week. On Day 1, 40 lunches are served. On Day 2, 85 lunches are served. On Day 3, 145 lunches are served. On Day 4, 220 lunches are served. The pattern continues.
Questions:
1. The number of lunches is increasing each day, and the increases are getting larger over time. From Day 1 to Day 2 it goes up a lot, and then it keeps going up even more each day.
2. No, the pattern is not increasing by the same amount each time. The increases are different each day (for example: +45, then +60, then +75), so the pattern is not constant.
Jessica has saved $50. She will add $25 to her savings each week. Ron has saved $40 and will add $25 to his savings each week. How much will each person have saved after 5 weeks?
Jessica will have $175 and Ron will have $165.
Maria put 3.25 into her savings account. Henry also put $2.50 into his savings account. Every week, they both put $0.50 into their accounts. What relationship do you notice between the amount Henry has saved and the amount Maria has saved each week?
Henry will always have $0.75 less than Maria
Sofia is planting flowers in rows for a garden. In the first row, she plants 4 flowers. In the second row, she plants 7 flowers. In the third row, she plants 10 flowers, and she continues the pattern the same way. Write the ordered pairs for the first five rows. THEN plot the points on a coordinate plane. What do you notice about the graph?
Table / Ordered pairs:
(1, 4), (2, 7), (3, 10), (4, 13), (5, 16)
Graph observation:
The points form a straight line that goes up steadily from left to right.
Mindy has already saved $20 and plans to save $8 each month. George has no money saved yet, but plans to save $5 each month. Will they ever have saved the same amount? Explain.
No, they will never have the same amount. The lines are getting further and further apart, so George will never catch up.
Annie and Kendall work at the library. At noon, Annie had reshelved 71 books, and Kendall had reshelved 57. Each of them shelves 30 more books each hour after noon.
How many books will each of them have shelved by 5 p.m?
Annie will have 207 books.
Ava is building a pattern with tiles for an art project. In Step 1, she uses 2 tiles. In Step 2, she uses 6 tiles. In Step 3, she uses 12 tiles. In Step 4, she uses 20 tiles. She continues the pattern in the same way.
Questions:
First five steps:
2, 6, 12, 20, 30
Pattern:
The increases are +4, +6, +8, +10.
The pattern is not increasing by the same amount each time—the increases are getting larger by 2 each time.
Obrien Landscapes is paying employees $20 plus $8 per lawn. Frankscapes company pays $35 plus $5 per lawn. Which company pays more? Explain.
Frankscapes pays more for fewer lawns, but Obrien Landscapes pays more once you do more lawns because it increases by $8 per lawn instead of $5.
Emma has $100 in her savings account. Jorge has $50 in his savings account. They each put $10 in their accounts at the end of each week. Complete the tables to see how much each of them has saved after 5 weeks. What pattern do you notice?
Emma will always have $50 more than Jorge
A box of breadcrumbs weighs 0.5 ounce and the breadcrumbs weigh 16 ounces. Can you identify any relationship between the weight of the boxes and the weight of the breadcrumbs?
The breadcrumbs weigh 32 times as much as the box
A movie theater is filling seats for a special show. In the first hour, 50 people are seated. By the second hour, 120 people are seated in total. By the third hour, 210 people are seated. By the fourth hour, 320 people are seated.
Questions:
Ordered pairs for the first five hours:
(1, 50), (2, 120), (3, 210), (4, 320), (5, 450)
The increases are +70, +90, +110, +130. The increase is not the same each time—it grows by 20 more each hour.
A science club is comparing how fast two water tanks are filling up.
Question:
Which tank has more water over time? Explain your reasoning.
Tank B starts with more water, so it is larger at the beginning. However, Tank A grows faster because it adds 7 liters each minute instead of 5. At 5 minutes, both tanks have the same amount of water (50 liters). After that, Tank A will have more water because it increases faster each minute.