Patterns and Algebra - Patterns

Use the table below to answer the question that follows. Which of the following equations correctly models the relationship in the table?

Which of the following equations correctly models the relationship in the table?

Mrs. Janie asked her class to write their own geometric sequence. As she walked around the room, she noticed Brad had 3,6,2,4,43,83,...3,6,2,4,34​,38​,...written on his paper. When Mrs. Janie asked Brad to explain his sequence, he replied, "My sequence is a geometric sequence because you multiply by 22, then multiply by 1331​, then you multiply by 22 again, then multiply by 1331​ again and so on." What is the best teacher response to Brad's statement?

"Your reasoning has an error. In order for a sequence to be a geometric sequence, you must multiply by the same common ratio each time. You can not use two different ratios."

Mr. White displayed 3 sequences on his whiteboard during math class: 20,10,5,0,−2.5,...20,10,5,0,−2.5,... 324,108,36,12,4,...324,108,36,12,4,... −6,1,8,15,22,...−6,1,8,15,22,... He asked his students to pick one of the sequences, determine if it was arithmetic or geometric, and explain their reasoning for their answer. Which student answer correctly identifies the sequence and has a sufficient explanation?

"−6,1,8,15,22,...−6,1,8,15,22,... is an arithmetic sequence because if you add 77 to each term, you get the value of the next term in the sequence. When you have a common difference, the sequence is arithmetic."

If the pattern below is continued, how many blocks will be in the 7th term?

36

Mrs. Fields' science class takes care of a plant at school. The class is responsible for watering the plant and measuring its growth. The students' measurements are charted in the table. Which of the following equations relates the height of the plant, h, and the number of months, m, assuming a constant rate of growth?

5m+3=h

What type of sequence is the following number sequence? 4, 12, 20, 28, 36, 44...

It is an arithmetic sequence; you add 8 every time to get the next term.

Students in Mrs. Portock's class were trying to write an expression that represents the total number of cubes for any term number, n, for the sequence below. Diego raised his hand to get help from Mrs. Portock. He had the table following written on his paper: Diego said he recognized the pattern of adding 3 more cubes every time, but needed help writing an expression using n. Of the following, what is the best possible teacher response?

Re-write the number of cubes as an expression showing the amount of cubes being added every time. (Ex: 1+3 instead of 4 cubes, 1+3+3 instead of 7 cubes, etc.)

Of the tables below, which one does not satisfy a linear equation?

This relationship has a constant second difference and is therefore not linear but quadratic. The equation �=�2−3y=x2−3 models this function.

What is the next shape in this pattern?

Triangle

A teacher prompted his class to write their own examples of arithmetic sequences on their papers. As the teacher is circulating, he sees the following written on one Tasha's paper: 2, 3, 5, 8, 12, 17, 23... When asked to explain her work, Tasha explains, " The sequence is arithmetic because to get the next number in the sequence, you +1, +2, +3, +4, +5, see? So you always add one more to get the next number." Which of the following is the best teacher response to Tasha's reasoning?

Your reasoning is incorrect; you must add by the same number every time for the sequence to be arithmetic.

The following sequence is an example of which type? 81,−27,9,−3,1,−13,...81,−27,9,−3,1,−31​,...

a geometric sequence, −13−31​ is used as the common ratio.

During small group instruction, a kindergarten teacher shows students a necklace with alternating red and blue beads. She asks students to share what they notice about the necklace. In this scenario, the teacher is most likely trying to promote:

an understanding of patterns.

What is the next shape in this pattern?..

circle

A preschool class works together with teacher guidance to separate various toys into three groups: small toys, medium toys, and large toys. Which of the following skills are students working on during this activity?

classifying objects

What is the next shape in this pattern?.

pentagon

Which equation best represents y in terms of x?

y = 1 - 2x

Kate is starting a new babysitting job in which she will make $7 per hour that she works. Currently, she owes her parents $15, which she will have to pay back from her earnings. Below is a table of the amount of money Kate has saved since starting her job. Which of the following equations could best be used to demonstrate the relationship between x and y?

y=7x−15