my homework lesson 3 sequences answer key

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How to Use a Math Medic Answer Key

Written by Luke Wilcox published 3 years ago

Answer key might be the wrong term here. Sure, the Math Medic answer keys do provide the correct answers to the questions for a lesson, but they have been carefully designed to do much more than this. They are meant to be the official guide to teaching the lesson, providing specific instructions for what to do and say to make a successful learning experience for your students.

Before we look at the details of the answer key, let's make sure we understand the instructional model first.

Experience First, Formalize Later (EFFL)

A typical Math Medic lesson always has the same four parts: Activity, Debrief Activity, QuickNotes, and Check Your Understanding. Here are the cliff notes:

Activity: Students are in groups of 2 - 4 working collaboratively through the questions in the Activity. The teacher is checking in with groups and using questions, prompts, and cues to get students to refine their communication and understanding. As groups finish the activity, the teacher asks students to go to the whiteboard to write up their answers to the questions.

Debrief Activity: In the whole group setting, the teacher leads a discussion about the student responses to the questions in the activity, often asking students to explain their thinking and reasoning about their answers. The teacher then formalizes the learning by highlighting key concepts and introducing new vocabulary, notation, and formulas.

QuickNotes: The teacher uses direct instruction to summarize the learning from the activity in the QuickNotes box - making direct connections to the learning targets for the lesson.

Check Your Understanding: Students are then asked to apply their learning from the lesson to a new context in the Check Your Understanding (CYU) problem. This can be done individually or in small groups. The CYU is very flexible in it's use, as it can be used as an exit ticket, a homework problem, or a quick review the next day.

How Do I See EFFL in the Answer Key?

You will see EFFL in the answer key like this:

Anything written in blue is something we expect our students to produce. This might not be quite what we expect by the end of the lesson, but provides us with a starting point when we move to formalization.

Anything written in red is an idea added by the teacher - the formalization of the learning that happened during the Activity. Students are expected to add these "notes" to their Activity using a red pen or marker.

What Do Students Write Down For Notes?

By the end of the lesson, students will have written down everything you see on the Math Medic Answer Keys. The most important transition is when students finish the Activity and we move to Debrief Activity. "Students, now is the time for you to put down your pencils and get out your your red Paper Mate flair pens" We give each student a Paper Mate flair pen at the beginning of the school year and tell them they must cherish and protect it with their life. They all think we should be sponsored by Paper Mate (anyone have any leads on this?)

The lessons you see on Math Medic are all of the notes we use with our students. We do not have some secret collection of guided notes.

Do Students Have Access to Answer Keys?

Yes! Any student can create a free Math Medic account to get access to the answer keys. We often send students to the website when they are absent from a lesson or when we don't quite finish the lesson in class. We are comfortable with students having access to these answer keys because we do not think Math Medic lessons should be used as a summative assessment or be used for a grade (unless it's for completion). Our lessons are meant to be the first steps in the formative process of learning new concepts.

Math Medic Help

Helping math teachers bring calculus to life

Change in Arithmetic Sequences (Lesson 4.1)

Unit 4 day 1 ced topic(s): 2.1, unit 4 day 1 day 2 day 3 day 4 day 5 day 6 day 7 day 8 day 9 day 10 day 11 day 12 day 13 day 14 day 15 all unit s, ​learning targets​.

Understand that sequences are a special type of function whose domain is the positive integers.

Write an explicit rule for arithmetic sequences using the common difference and any term in the sequence.

Apply understanding of how arithmetic sequences grow to determine the common difference, find missing terms and reason about arithmetic sums.

Quick Lesson Plan

Activity: #goals.

Lesson Handout

Experience first:.

In today’s activity, students look at Mallory’s training schedule during the month of July to explore the idea of arithmetic sequences. Students identify that her time increases by five minutes every day and use this to fill in her running log. While students may use a recursive pattern to find the first few values in the table, they should quickly recognize the need to make use of structure to find values for days later in July. We specifically ask for July 30th so students recognize that her running time on that day is exactly five minutes less than her running time on the 31st. This idea of a constant (common) difference is critical to the rest of this lesson and both reviews and previews ideas about constant rate of change and linear functions. We also specifically ask for her running time on July 16th since this marks the exact midpoint of the month. Students may use this value in questions 3 and 4. Students may also average the first and last term directly as a way to find her average run time. Be prepared that students may also try to add all 31 terms and divide by 31. 

Questions 5 and 6 are all about getting students to see that any term in the sequence can be used as an “anchor point” for the sequence. Instead of using repeated subtraction to find an initial term and then repeated addition to find the desired term, it is intuitive to simply add on or subtract from the given term. The debrief should focus on how students decided how many “copies” of 0.2 to add or subtract. This of course is the (n-k) portion of the explicit formula and represents how many terms the desired term is away from the given term.

Monit oring Questions:

How many more minutes will Mallory run 3 days from now?

What’s the average of the numbers 5, 6, 7, 8, and 9? What’s the average of the numbers 3, 5, 7, 9, 11?

How many days have passed since July 18th? Why does that matter?

Is David going to run more or less than 7.6 miles on this day?

Formalize Later:

In this unit we focus on functions exhibiting exponential growth or decay. That of course begins with a conversation about the rate of change of exponential functions and to study this, we compare and contrast linear and exponential functions. However, we back up even further and begin by studying change in arithmetic and geometric sequences which are special types of functions whose domain is the positive integers. The focus on absolute change in sequences (how do the terms change from one to the next?) is a great introduction to the rate of change in more general functions (how do the outputs change with respect to the inputs?)

Lesson 4.1 focuses on arithmetic sequences, or sequences with a common difference. One important goal of the lesson is to show students that explicit rules can be written given any term in the sequence, not just the 0th term or the 1st term. This is analogous to point-slope form for linear functions.

Though the AP Precalculus course framework does not explicitly mention partial sums of sequences, the Calc Medic curriculum does address them informally for the purpose of differentiating the features of arithmetic and geometric sequences. Knowing how a sequence grows is paramount to determining a strategy for finding a partial sum. In the next lesson, we do this in reverse. Knowing how geometric sequences grow prohibits us from using the same strategy as we did for arithmetic sequences to find a partial sum (i.e. finding the “middle” value and multiplying by the number of terms). Note that we will not ask students explicitly to find the nth partial sum or use the partial sum notation.

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13.3E: Geometric Sequences (Exercises)

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13. Find the common ratio for the geometric sequence \(2.5, \quad 5, \quad 10, \quad 20, \ldots\)

14. Is the sequence \(4,16,28,40 \ldots\) geometric? If so find the common ratio. If not, explain why.

15. A geometric sequence has terms \(a_{7}=16,384\) and \(a_{9}=262,144 .\) What are the first five terms?

16. A geometric sequence has the first term \(a_{1}=-3\) and common ratio \(r=\frac{1}{2} .\) What is the \(8^{\text {th }}\) term?

17. What are the first five terms of the geometric sequence \(a_{1}=3, \quad a_{n}=4 \cdot a_{n-1} ?\)

18. Write a recursive formula for the geometric sequence \(1, \quad \frac{1}{3}, \quad \frac{1}{9}, \quad \frac{1}{27}, \ldots\)

19. Write an explicit formula for the geometric sequence \(-\frac{1}{5}, \quad-\frac{1}{15}, \quad-\frac{1}{45}, \quad-\frac{1}{135}, \ldots\)

20. How many terms are in the finite geometric sequence \(-5,-\frac{5}{3},-\frac{5}{9}, \ldots,-\frac{5}{59,049} ?\)

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Eureka Math Algebra 1 Module 3 Lesson 3 Answer Key

Engage ny eureka math algebra 1 module 3 lesson 3 answer key, eureka math algebra 1 module 3 lesson 3 exercise answer key.

Exercise 2. Think of a real – world example of an arithmetic or a geometric sequence. Describe it, and write its formula. Answer: Answers will vary. An example of an arithmetic sequence would be a person’s salary that increases by $2,000 each year. A recursive formula would be S(n + 1) = S(n) + 2,000 for n ≥ 1 for some initial salary S(1). An example of a geometric sequence would be a person’s salary that increases by 2% each year. A recursive formula would be S(n + 1) = 1.02S(n) for n ≥ 1 for some initial salary S(1).

Exercise 3. If we fold a rectangular piece of paper in half multiple times and count the number of rectangles created, what type of sequence are we creating? Can you write the formula? Answer: We are creating a geometric sequence because each time we fold, we double the number of rectangles. R(n) = 2 n , where n is the number of times we have folded the paper.

Eureka Math Algebra 1 Module 3 Lesson 3 Problem Set Answer Key

For Problems 1–4, list the first five terms of each sequence, and identify them as arithmetic or geometric. Question 1. A(n + 1) = A(n) + 4 for n ≥ 1 and A(1) = – 2 Answer: – 2, 2, 6, 10, 14 Arithmetic

Question 2. A(n + 1) = \(\frac{1}{4}\) ⋅ A(n) for n ≥ 1 and A(1) = 8 Answer: 8, 2, \(\frac{1}{2}\), \(\frac{1}{8}\), \(\frac{1}{32}\) Geometric

Question 3. A(n + 1) = A(n) – 19 for n ≥ 1 and A(1) = – 6 Answer: – 6, – 25, – 44, – 63, – 82 Arithmetic

Question 4. A(n + 1) = \(\frac{2}{3}\) A(n) for n ≥ 1 and A(1) = 6 Answer: 6, 4, \(\frac{8}{3}\), \(\frac{16}{9}\), \(\frac{32}{27}\) Geometric

For Problems 5–8, identify the sequence as arithmetic or geometric, and write a recursive formula for the sequence. Be sure to identify your starting value. Question 5. 14, 21, 28, 35, … Answer: f(n + 1) = f(n) + 7 for n ≥ 1 and f(1) = 14 Arithmetic

Question 6. 4, 40, 400, 4000, … Answer: f(n + 1) = 10f(n) for n ≥ 1 and f(1) = 4 Geometric

Question 7. 49, 7, 1, \(\frac{1}{7}\), \(\frac{1}{49}\), … Answer: f(n + 1) = \(\frac{1}{7}\)f(n) for n ≥ 1 and f(1) = 49 Geometric

Question 8. – 101, – 91, – 81, – 71, … Answer: f(n + 1) = f(n) + 10 for n ≥ 1 and f(1) = – 101 Arithmetic

Question 9. The local football team won the championship several years ago, and since then, ticket prices have been increasing $20 per year. The year they won the championship, tickets were $50. Write a recursive formula for a sequence that models ticket prices. Is the sequence arithmetic or geometric? Answer: T(n) = 50 + 20n, where n is the number of years since they won the championship; n ≥ 1 (n ≥ 0 is also acceptable). The sequence is arithmetic. OR T(n + 1) = T(n) + 20, where n is the number of years since the year they won the championship; n ≥ 1 and T(1) = 70 (n ≥ 0 and T(0) = 50 is also acceptable). The sequence is arithmetic.

Question 10. A radioactive substance decreases in the amount of grams by one – third each year. If the starting amount of the substance in a rock is 1,452 g, write a recursive formula for a sequence that models the amount of the substance left after the end of each year. Is the sequence arithmetic or geometric? Answer: A(n + 1) = \(\frac{2}{3}\) A(n) or A(n + 1) = 2A(n)÷3, where n is the number of years since the measurement started, A(0) = 1,452 The sequence is geometric. Since the problem asked how much radioactive substance was left, students must take the original amount, divide by 3 or multiply by \(\frac{1}{3}\), and then subtract that portion from the original amount. An easier way to do this is to just multiply by the amount remaining. If \(\frac{1}{3}\) is eliminated, \(\frac{2}{3}\) remains.

Question 11. Find an explicit form f(n) for each of the following arithmetic sequences (assume a is some real number and x is some real number). a. – 34, – 22, – 10, 2, … Answer: f(n) = – 34 + 12(n – 1) = 12n – 46, where n ≥ 1

b. \(\frac{1}{5}\), \(\frac{1}{10}\), 0, – \(\frac{1}{10}\), … Answer: f(n) = \(\frac{1}{5}\) – \(\frac{1}{10}\)(n – 1) = \(\frac{3}{10}\) – \(\frac{1}{10}\)n, where n ≥ 1

c. x + 4, x + 8, x + 12, x + 16, … Answer: f(n) = x + 4 + 4(n – 1) = x + 4n, where n ≥ 1

d. a, 2a + 1, 3a + 2, 4a + 3, … Answer: f(n) = a + (a + 1)(n – 1) = a + an – a + n – 1 = an + n – 1, where n ≥ 1

Question 12. Consider the arithmetic sequence 13, 24, 35, …. a. Find an explicit form for the sequence in terms of n. Answer: f(n) = 13 + 11(n – 1) = 11n + 2, where n ≥ 1

b. Find the 40th term. Answer: f(40) = 442

c. If the nth term is 299, find the value of n. Answer: 299 = 11n + 2 → n = 27

Question 13. If – 2, a, b, c, 14 forms an arithmetic sequence, find the values of a, b, and c. Answer: 14 = – 2 + (5 – 1)d 16 = 4d d = 4

a = – 2 + 4 = 2 b = 2 + 4 = 6 c = 6 + 4 = 10

Question 14. 3 + x, 9 + 3x, 13 + 4x, … is an arithmetic sequence for some real number x. a. Find the value of x. Answer: The difference between term 1 and term 2 can be expressed as (9 + 3x) – (3 + x) = 6 + 2x. The difference between term 2 and term 3 can be expressed as (13 + 4x) – (9 + 3x) = 4 + x. Since the sequence is known to be arithmetic, the difference between term 1 and term 2 must be equal to the difference between term 2 and term 3. Thus, 6 + 2x = 4 + x, and x = – 2; therefore, the sequence is 1, 3, 5, ….

b. Find the 10th term of the sequence. Answer: f(n) = 1 + 2(n – 1) = 2n – 1, where n ≥ 1 f(10) = 19

Question 15. Find an explicit form f(n) of the arithmetic sequence where the 2nd term is 25 and the sum of the 3rd term and 4th term is 86. Answer: a,25,b,c 25 = a + (2 – 1)d 25 = a + d

b = 25 + d b = a + 2d

c = 25 + 2d c = a + 3d b + c = (a + 2d) + (a + 3d) = 2a + 5d = 86 a + d = 25 Solving this system: d = 12, a = 13, so f(n) = 13 + 12(n – 1), where n ≥ 1 b = 13 + 2(12) = 37 c = 13 + 3(12) = 49 OR b + c = (25 + d) + (25 + 2d) = 50 + 3d = 86 → d = 12 25 = a + 12 → a = 13 b = 25 + 12 → b = 37 c = 25 + 2(12) → c = 49 So, f(n) = 13 + 12(n – 1).

Question 17. Find the common ratio and an explicit form in each of the following geometric sequences. a. 4, 12, 36, 108, … Answer: r = 3 f(n) = 4(3) (n – 1) , where n ≥ 1

b. 162, 108, 72, 48, … Answer: r = \(\frac{108}{162}\) = \(\frac{2}{3}\) f(n) = 162(\(\frac{2}{3}\)) (n – 1) , where n ≥ 1

c. \(\frac{4}{3}\), \(\frac{2}{3}\), \(\frac{1}{3}\), \(\frac{1}{6}\), … Answer: r = \(\frac{1}{2}\) f(n) = (\(\frac{4}{3}\)) (\(\frac{1}{2}\)) (n – 1) = (\(\frac{4}{3}\)) (2) (1 – n) , where n ≥ 1

d. xz, x 2 z 3 , x 3 z 5 , x 4 z 7 , … Answer: r = xz 2 f(n) = xz(xz 2 ) (n – 1) , where n ≥ 1

Question 18. The first term in a geometric sequence is 54, and the 5th term is 2/3. Find an explicit form for the geometric sequence. Answer: \(\frac{2}{3}\) = 54(r) 4 \(\frac{1}{81}\) = r 4 r = \(\frac{1}{3}\) or – \(\frac{1}{3}\) f(n) = 54(\(\frac{1}{3}\)) (n – 1)

Question 19. If 2, a, b, – 54 forms a geometric sequence, find the values of a and b. Answer: a = 2r b = 2(r) 2 – 54 = 2(r) 3 – 27 = r 3 – 3 = r, so a = – 6 and b = 18

Question 20. Find the explicit form f(n) of a geometric sequence if f(3) – f(1) = 48 and \(\frac{f(3)}{f(1)}\) = 9. Answer: f(3) = f(1) (r) 2 \(\frac{f(3)}{f(1)}\) = r 2 = 9 r = 3 or – 3 f(1) r 2 – f(1) = 48 f(1)(r 2 – 1) = 48 f(1)(8) = 48 f(1) = 6 f(n) = 6(3) (n – 1) , where n ≥ 1 or f(n) = 6( – 3) (n – 1) , where n ≥ 1

Eureka Math Algebra 1 Module 3 Lesson 3 Exit Ticket Answer Key

Question 1. Write the first three terms in the following sequences. Identify them as arithmetic or geometric. a. A(n + 1) = A(n) – 5 for n ≥ 1 and A(1) = 9 Answer: 9, 4, – 1 Arithmetic

b. A(n + 1) = \(\frac{1}{2}\) A(n) for n ≥ 1 and A(1) = 4 Answer: 4, 2, 1 Geometric

c. A(n + 1) = A(n)÷10 for n ≥ 1 and A(1) = 10 Answer: 10, 1, \(\frac{1}{10}\) or 10, 1, 0.1 Geometric

Question 2. Identify each sequence as arithmetic or geometric. Explain your answer, and write an explicit formula for the sequence. a. 14, 11, 8, 5, … Answer: Arithmetic – 3 pattern 17 – 3n, where n starts at 1

b. 2, 10, 50, 250, … Answer: Geometric ×5 pattern 2(5 (n – 1) ), where n starts at 1

c. – \(\frac{1}{2}\), – \(\frac{3}{2}\), – \(\frac{5}{2}\), – \(\frac{7}{2}\), … Answer: Arithmetic – 1 pattern \(\frac{1}{2}\) – n, where n starts at 1

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