few_shots: 
Example 1:
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?


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.

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:
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 to multiply a fraction or whole number by a fraction.

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

Example 3:
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.
{"table": [{"_attributes": {"align": "center", "class": ["table", "table-bordered"], "style": "table-layout:fixed; width:70%"}, "tbody": [{"tr": [{"td": [{"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "No solution"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Exactly 1 solution"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Infinite number of solutions"}]}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9=4x+7}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${y=3y+5}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${y={x\\over2}}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${3x+6y-9=0}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${x=y}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${1.25x=1.25y}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9.9=6x+8y}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9x=2.5y-8.8}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${11x-2y=1.5}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${11x-2y=2.5}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {}]}]}]}]}
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?


Concept/skill: 
Analyze and solve pairs of simultaneous linear equations.

Answer: 
Yes. The last three rows of the table in the first problem involves solving pairs of simultaneous linear equations. The second word problem can also be represented as two simultaneous linear equations.

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_000972_pos_0

6

fl_problem_001097_neg_0

7

fl_problem_001392_pos_0

8
standard_description: Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
problem_activity: Activity
Students now explore the idea of repeated percent change in a banking or savings context. This may be a good time to more formally introduce the term
compounding
. They examine the effect of compounding 1% interest each month for a year. From the previous activity, students should recognize that the result will be
more
than 12% interest on the initial balance. This is because for each month after the first one, the amount used to calculate the interest includes previously earned interest, so that amount is higher than the initial balance. In future lessons, students will continue to investigate the effects of compounding interest in the long term.
To find the account balance at different times (the first question), students may choose to calculate the numerical values or to write expressions. Those who choose the latter will be in a better position to write a generalized expression for the account balance at any time. Select some students who chose different approaches to share their choice later.
Note that this activity simplifies what banks actually do to calculate interests. Financial institutions commonly calculate interest rates based on the number of days in the month or the year. The monthly interest rate for February, for instance, will be lower than that for March.
Launch
Give students a brief overview of savings accounts. Explain that when we deposit money in a savings account, we are allowing the bank to hold on to the money (and potentially use it to generate more revenue for the bank). In exchange, the bank pays us interest, that is a certain percentage of the money at a regular interval. The more money we put in the account, the more interest we earn. The interest calculation here works the same way as that for loans and credit cards, except that, in this case, higher interest rates and more-frequent interest calculations work in our favor, instead of against us.
Arrange students in groups of 2.
Representation: Internalize Comprehension.
Activate or supply background knowledge about generalizing a process to create an expression or equation. After students calculate the account balance after 6 months, 1 year, 2 years, and 5 years, pause and check to make sure students are using the correct units of time (months instead of years). Invite students to identify which parts of their process changed and which stayed the same, and to use this information to create an equation for the account balance (
\(a\)
) in terms of the number of months (
\(m\)
).
Supports accessibility for: Visual-spatial processing; Conceptual processing
Student Facing
A bank account has a monthly interest rate of 1% and initial balance of
$
1,000. Any earned interest is added to the account and no other deposits or withdrawals are made.
What is the account balance after 6 months, 1 year, 2 years, and 5 years? Show your reasoning.
Write an equation expressing the account balance (
\(a\)
) in terms of the number of months (
\(m\)
). Assume that all interest earned continues to be added to the account and no other deposits or withdrawals are made.
How much interest will the account earn in 1 year? What percentage of the initial balance is that? Show your reasoning.
The term annual return refers to the percent of interest an account holder could expect to receive in one year. Discuss with your partner: If you were the bank, would you advertise the account as having a 12% annual return? Why or why not? Use your work so far to explain your reasoning.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Students may miss that the interest is applied monthly rather than annually. Help them read the question carefully, note this difference, and adjust the units to months for the appropriate questions.
Activity Synthesis
Ask previously selected students to share their strategies for the first question (using numerical values or using expressions). Highlight that writing expressions enables us to generalize and find the account balance for any time frame.
Select students to share how they found the interest earned over one year and what percentage it was of the initial balance. Make sure students understand that twelve successive increases of 1% yield a balance that is more than a single 12% increase. Then, invite groups to share which annual interest rate they would advertise if they were the bank.
Conclude by explaining that banking institutions call the two different interest rates different names. The 12% is called the
nominal rate
and the 12.68% the
effective rate
. The latter reflects the actual interest amount earned over a year, taking into account the monthly interest payment. This process of generating new earnings from previous earnings is called
compounding.
(In the context of loans,
compounding
refers to the generation of new debt from previous debts.)
As in the previous activity, the difference between the one-year interest calculated using the nominal rate versus the actual rate is not large. The differences become more and more pronounced, however, as time goes on.
Conversing: MLR2 Collect and Display.
During the discussion, listen for and collect the language students use to describe the situation involving a bank account that earns monthly interest. Call students’ attention to language such as “initial or starting balance,” “annual return,” or “initial earning plus the interest.” Write the students’ words and phrases on a visual display and update it throughout the remainder of the lesson. Be sure that the display includes new terms such as “nominal rate,” “effective rate,” and “compounding.” Remind students to borrow language from the display as needed. This will help students use mathematical language for describing a repeated percent change—compounding—in a banking context.
Design Principle(s): Maximize meta-awareness


aligns: no