few_shots: 
Example:
Problem/activity:
For each pair of numbers, determine if the values are equivalent. Explain your reasoning for each one.
a.
$$-0.25$$
and
$$\frac{1}{-4}$$
b.
$$-\left ( \frac{1}{4} \right )$$
and
$$\frac{-1}{-4}$$
c.
$$\frac{-1}{4}$$
and
$$-\frac{1}{4}$$


Concept/skill: 
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

Answer: 
Yes. This problem involves understanding fractions and division to divide rational numbers, such as 1, -4, 4, and -1. This problem also involves an understanding of multiplication: if p and q are integers, then -(p/q) = (-p)/q = p/(-q).

Now, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_001097_pos_0

7
standard_description: Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., p^(2)). For example, by truncating the decimal expansion of Ö2, show that Ö2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
problem_activity: Problem 1
Here are two algorithms for finding the value of 64 × 2.
|                                  Patrice’s method                                   |                                                                                                             Jerome’s method                                                                                                             |
|-------------------------------------------------------------------------------------|-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
| .. image:: https://d21royfkw9g4l6.cloudfront.net/uploads/2021/08/26/4-2-10-at1a.png | |https://d21royfkw9g4l6.cloudfront.net/uploads/2021/08/26/4-2-10-at1b.png|  |https://d21royfkw9g4l6.cloudfront.net/uploads/2021/08/26/4-2-10-at1b.png| image:: https://d21royfkw9g4l6.cloudfront.net/uploads/2021/08/26/4-2-10-at1b.png |
a.   How is Patrice’s method similar to and different from Jerome’s method?
b.   How do you think Jerome found the product 128?
Problem 2
Patrice solved the problem 813 × 7 using the partial products algorithm, as shown below:
[IMG]
Jerome solved in a different way from Patrice. Here is what he did:
|                      First, he found the partial product of 3 ones and 7, which is 21 ones, or 2 tens 1 one. He wrote the ones digit, 1, under the equal bar, and the 2 tens on the equal bar above the tens place in smaller handwriting.                      | Then, he found the partial product of 1 ten and 7, which was 7 tens. Because he had the 2 tens from the first partial product, he added those tens together and wrote 9 in the tens place in the product. Because he had accounted for the 2 tens from his first partial product, he crossed it off. |                                                  Finally, he found the partial product of 8 hundreds and 7, which was 56 tens. He wrote 56 in the hundreds and tens place in the product.                                                   |
|-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
| |https://d21royfkw9g4l6.cloudfront.net/uploads/2021/08/26/4-2-10-at2b_bGxguvo.png|  |https://d21royfkw9g4l6.cloudfront.net/uploads/2021/08/26/4-2-10-at2b_bGxguvo.png| image:: https://d21royfkw9g4l6.cloudfront.net/uploads/2021/08/26/4-2-10-at2b_bGxguvo.png |                               |https://d21royfkw9g4l6.cloudfront.net/uploads/2021/08/26/4-2-10-at2c.png|  |https://d21royfkw9g4l6.cloudfront.net/uploads/2021/08/26/4-2-10-at2c.png| image:: https://d21royfkw9g4l6.cloudfront.net/uploads/2021/08/26/4-2-10-at2c.png                                | |https://d21royfkw9g4l6.cloudfront.net/uploads/2021/08/26/4-2-10-at2d.png| * *  |https://d21royfkw9g4l6.cloudfront.net/uploads/2021/08/26/4-2-10-at2d.png| image:: https://d21royfkw9g4l6.cloudfront.net/uploads/2021/08/26/4-2-10-at2d.png |
a.   How is Patrice's method similar to and different from Jerome's method?
b.   What are the advantages and disadvantages of Patrice's method? Of Jerome's method?
Problem 3
Try using the standard algorithm to solve. If you get stuck, use an area model and/or the partial products algorithm to help. Then assess the reasonableness of your answer.
a.
$$54\times6$$
b.
$$9\times673$$
c.
$$1,605\times4$$


aligns: no