Unit 1 Lesson 1 Practice Problems IM® Algebra 2TM authored by Illustrative Mathematics®

Summary

The transcript discusses generating sequences of numbers based on specific rules. The first sequence starts with 0 and 1, and each subsequent number is the sum of the previous two, resulting in a Fibonacci-like sequence. The second sequence begins with fractions, doubling the denominator with each term. The third sequence involves multiplying by -2 and adding 3, and the process is reversed to find terms before a given number. Lastly, a sequence starting with 0 and 5 is explored, with rules of adding 5 or multiplying by 2 and adding 5 to generate further terms.

Takeaways

• 🔢 The first sequence is generated by starting with 0 and 1, then each subsequent number is the sum of the previous two, resulting in the sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34.
• 📉 The second sequence starts with fractions 1/2, 1/4, 1/8, and follows a rule where each term is the previous term multiplied by 1/2 or the denominator is doubled.
• 🔄 To find the next three terms in the second sequence, the rule is to take the previous term, multiply by 1/2, and then double the denominator, resulting in 1/16, 1/32, 1/64.
• ➗ For the third sequence starting with -7, the rule involves multiplying the previous number by -2 and then adding 3, leading to the next three terms being 17, -31, and 67.
• 🔙 To find the three terms before -7 in the third sequence, reverse the rule by subtracting 3 and then dividing by -2, resulting in the terms 5, -1, and 2 before -7.
• 🔄 The fourth sequence starts with 0 and follows a rule where each term is the previous term plus 5, giving the next three terms as 10, 15, and 20.
• 🔢 An alternative rule for the fourth sequence could be to multiply the previous term by 2 and then add 5, leading to the next three terms being 15, 35, and 75.
• 🤔 The script suggests that there can be multiple valid rules for generating sequences, and the specific rule chosen can lead to different sequences.
• 🔍 The script emphasizes the importance of being able to reverse the rules to check the accuracy of the generated sequences by working backwards.

Q & A

What is the rule for generating the third number in the first sequence starting with 0 and 1?

The rule is to take the sum of the previous two numbers in the sequence. So, the third number is 0 + 1, which equals 1.

How do you calculate the fifth number in the first sequence?

Following the rule of summing the previous two numbers, the fifth number is calculated as 2 (the third number) + 3 (the fourth number), which equals 5.

What is the 10th number in the sequence that starts with 0 and 1?

The 10th number in the sequence is found by adding the 8th and 9th numbers: 13 (the 8th number) + 21 (the 9th number), which equals 34.

What rule describes the pattern in the second sequence starting with 1/2, 1/4, and 1/8?

The rule for the second sequence is to multiply the previous term's denominator by 2, keeping the numerator constant at 1.

How do you find the next term in the second sequence after 1/8?

To find the next term after 1/8, you double the denominator, resulting in 1/16.

What is the rule for the third sequence that starts with the term -7?

The rule for the third sequence is to multiply the previous number by -2 and then add 3.

What are the next three terms after -7 in the third sequence?

Following the rule, the next three terms after -7 are 17 (-7 * -2 + 3), -31 (17 * -2 + 3), and 67 (-31 * -2 + 3).

How can you find the term before -7 in the third sequence by working backwards?

To find the term before -7, you subtract 3 and then divide by -2, which gives you 5 (-7 - 3) / -2.

What is the term that comes before 5 in the sequence when working backwards?

Working backwards from 5, you subtract 3 to get 2, and then divide by -2 to get -1 (2 / -2).

What are three possible terms that could come before -7 in the third sequence?

Three possible terms that could come before -7, when working backwards, are 5 (-7 - 3) / -2, 2 (5 - 3) / -2, and -1 (2 - 3) / -2.

What rule could be applied to generate the next three terms after 0 and 5 in the fourth sequence?

One possible rule for the fourth sequence could be to add 5 to the previous term, resulting in the next three terms being 10 (5 + 5), 15 (10 + 5), and 20 (15 + 5).

How do you generate the next three terms in the fourth sequence using a different rule?

Using a different rule, such as multiplying the previous term by 2 and then adding 5, the next three terms after 5 would be 15 (0 * 2 + 5), 35 (15 * 2 + 5), and 75 (35 * 2 + 5).