few_shots: 
Example 1:
Concept/skill: 
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

Problem/activity:
You’re mixing ingredients for cookies. The recipe says to combine 6 tablespoons, or ¾ stick, of butter with 1 cup of sugar. You accidentally mix in a whole stick of butter (8 tablespoons) with the cup of sugar. How can you fix this?


Answer: 
Yes. This problem solves a real-world problem involving mixing ingredients for cookies. A student doing this problem would need to reason about equivalent ratios of butter amounts to sugar amounts.

Example 2:
Concept/skill: 
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

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
$$-\frac{1}{4}$$


Answer: 
No. The expressions shown in this problem do not involve multiplying fractions with other fractions, nor whole numbers with fractions.

Example 3:
Concept/skill: 
Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

Problem/activity:
Problem 1
The table below contains single equations in a single variable, single equations in two variables, and pairs of simultaneous equations in two variables. Check the appropriate box to show whether there is no solution, exactly 1 solution, or if there is an infinite number of solutions.
|                                                          | **           No solution          ** | **           Exactly 1 solution          ** | **           Infinite number of solutions          ** |
|----------------------------------------------------------|--------------------------------------|---------------------------------------------|-------------------------------------------------------|
|                       $${9=4x+7}$$                       |                                      |                                             |                                                       |
|                       $${y=3y+5}$$                       |                                      |                                             |                                                       |
|                    $${y={x\over2}}$$                     |                                      |                                             |                                                       |
|                     $${3x+6y-9=0}$$                      |                                      |                                             |                                                       |
|    $${x=y}$$                        $${1.25x=1.25y}$$    |                                      |                                             |                                                       |
| $${9.9=6x+8y}$$                        $${9x=2.5y-8.8}$$ |                                      |                                             |                                                       |
| $${11x-2y=1.5}$$                        $${11x-2y=2.5}$$ |                                      |                                             |                                                       |
Problem 2
Kim has a small container and a large container as shown.
[IMG]
It takes 16 of the small containers to fill the large container. Three small containers leave 1.95 gallons of space in the big container.
What is the size of each of the two containers?


Answer: 
No. Completing neither of these problems requires replacing one equation by the sum of that equation and a multiple of the other equation.

Now, determine whether the following problem or activity can enable students to understand the concepts or skills in the provided description.

shot_info: fl_problem_000972_pos_0

6

fl_problem_001097_neg_0

7

fl_problem_001392_neg_0

8
standard_description: Know and apply the Binomial Theorem for the expansion of (x + y)^(n )in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle. (The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.)
problem_activity: Narrative
This activity prompts students to examine patterns in multiples of 10 and 9, and to notice that the digits in the multiples of 9 can be reasoned in relation to the more-familiar multiples of 10. Students use what they know about the place value and operations to explain the patterns in these multiples (MP7). For instance, students may reason that, because 9 is 1 less than 10, to find
\(12 \times 9\)
is to find the
\(12 \times 10\)
and then subtract 1 twelve times (or subtract
\(12 \times 1\)
) from the product.
The reasoning in this activity prepares them to notice patterns in the multiples of 100 and 99 in the next lesson.
MLR8 Discussion Supports.
Use multimodal examples to show the patterns of both columns. Use verbal descriptions along with gestures, drawings, or concrete objects to show the connection between the multiples of 9 and 10.
Advances: Listening, Representing
Representation: Access for Perception.
Synthesis: Use pictures of the long rectangle base-ten blocks to help students visualize the patterns. For example, display a picture of eight long rectangle base-ten blocks. Count by 10 while pointing to each block. Then, cross out one unit in each block and discuss how this shows that counting by 9 is like multiplying by 10 and subtracting.
Supports accessibility for: Conceptual Processing, Visual Spatial Processing
Launch
Groups of 2
Read the opening paragraph as a class.
“Which do you prefer, counting by 10 or counting by 9? Why?” (Counting by 10 because I've been doing that since kindergarten.)
30 seconds: partner discussion
“Let’s look at numbers we get by counting by 9 and by 10 and see what patterns we can find.”
Activity
“Take a few quiet minutes to work on the first few problems. Then, share your thinking and complete the rest of the activity with your partner.”
5–6 minutes: independent work time
5–6 minutes: partner discussion
Monitor for students who:
can clearly explain the pattern of the digits in multiples of 10 in terms of place value
notice connections between the values in the two columns and use them to explain the patterns in the digits in multiples of 9
Student Facing
Andre’s class is choral counting by 10 and then by 9. The left column shows the numbers they say when counting by 10.
Complete the right column with the first ten numbers the class will say when counting by 9.
What patterns do you notice about the features of the numerical patterns? Make at least two observations about each list of numbers.
| counting by 10 | counting by 9 |
|----------------|---------------|
|       10       |               |
|       20       |               |
|       30       |               |
|       40       |               |
|       50       |               |
|       60       |               |
|       70       |               |
|       80       |               |
|       90       |               |
|      100       |               |
For the numbers in the “counting by 10” column, why do you think:
the digits in the tens place change the way they do?
the digits in the ones place are the way they are?
For the numbers in the “counting by 9” column, why do you think the digits in the ones place change the way they do? Explain your reasoning.
Student Response
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Advancing Student Thinking
Students may not see that the digit in the ones place decreases by 1 each time the count goes up by 9. Consider asking:
“What patterns do you see in the pair of numbers in each row?”
“How do you think the numbers in the ‘counting by 9’ column are related to those in the ‘counting by 10’ column?”
Activity Synthesis
Display the completed table.
Invite students to share the patterns they noticed in the numbers in each column. Record their observations by annotating the numbers in the table.
If no students mentioned that the two sets of numbers are multiples of 10 and multiples of 9, ask them about it.
Select previously identified students to share their responses to the last two problems.
If no students reason that counting by 9 can be thought of as counting by 10 and subtracting 1 each time, bring this to their attention.
In other words, counting by 9 once means
\(10- 1\)
, which is 9. Counting by 9 again means adding another
\(10 -1\)
to 9, or
\(9 + 10 - 1\)
, which is 18.  Counting by 9 a third time means
\(18 + 10 - 1\)
, which is 27. And so on.
“Counting by 9 eight times is the same as counting by 10 eight times and subtracting 1 eight times, or
\((8 \times 10) - (8 \times 1)\)
, which is
\(80 - 8\)
or 72.”


aligns: no