few_shots: 
Example 1:
Problem/activity:
Task
Materials
One 10-frame per pair of students (see example below)
[IMG]
Counters, 10 per pair of students
Decks of number cards 0-5 (with 4 of each number) or two 0-5 dice
Setup
This activity is done in pairs. The students sit side by side with 10-frame and deck of cards or dice. If they have cards, the cards are shuffled and placed face down in front of them.
Action
Player 1 flips a card or rolls the die. They place that many counters on the 10-frame. Player 2 flips a card or rolls the die. They place that many more counters on the 10-frame next to the counters already on the ten frame. The students count the total together.
A recording sheet can be added when the students have been introduced to equations and number sentences. For those students who can handle higher numbers, a deck of 0-10 cards and two 10-frames can be used.
This game can also be played alone.


Concept/skill: 
Represent addition and subtraction with objects, fingers, mental images, drawings (Drawings need not show details, but should show the mathematics in the problem. (This applies wherever drawings are mentioned in the Standards.)), sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.

Answer: 
Yes. This task involves representing addition with objects such as number cards and dice.

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:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Answer: 
Yes. An initial step when solving for x in this problem requires the use of the addition formula for cosine on the left side of the equation.

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

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standard_description: Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.
problem_activity: Activity
The purpose of this activity is to bring together identities and the work students did earlier with rational equations. For each question, students first rewrite the fraction
\(\frac{2}{15}\)
using the given formula and then prove that the formula is an identity.
The whole-class discussion at the end of the activity should focus on how students showed that the formula is an identity. Monitor for groups who use different methods to share during the discussion. For example, they may identify different common denominators when adding together the fractions in the expression on the right side of the formula.
Launch
Arrange students in groups of 2. Tell students to complete each problem individually, and then compare their work with their partner. Partners should discuss any similarities and differences in their methods, and decide which method they prefer.
If students need some practice adding fractions with unlike denominators, begin the activity by displaying the equation
\(\frac78 = \frac12 + \frac13 + \frac{1}{24}\)
for all to see and ask students how they can figure out without using a calculator that the equation is true. After some quiet work time, select 1–2 students to share their thinking about why the equation is true. The purpose of looking at this equation before the main activity is to make sure students recall that adding fractions requires a common denominator.
Speaking: MLR8 Discussion Supports.
Use this routine to support students in producing statements justifying their identity formulas. Provide sentence frames for students to use when comparing their methods and discussing which method they prefer, such as “I prefer _____ because _____ .”, or “We both have _____ which represents _____ .”
Design Principle(s): Support sense-making
Student Facing
[IMG]
In Ancient Egypt, all non-unit fractions were represented as a sum of distinct unit fractions. For example,
\(\frac49\)
would have been written as
\(\frac13 + \frac19\)
(and not as
\(\frac19+\frac19+\frac19+\frac19\)
or any other form with the same unit fraction used more than once). Let’s look at some different ways we can rewrite
\(\frac{2}{15}\)
as the sum of distinct unit fractions.
Use the formula
\(\frac{2}{d}=\frac{1}{d}+\frac{1}{2d}+\frac{1}{3d}+\frac{1}{6d}\)
to rewrite the fraction
\(\frac{2}{15}\)
, then show that this formula is an identity.
Another way to rewrite fractions of the form
\(\frac{2}{d}\)
is given by the identity
\(\frac{2}{d} = \frac{1}{d} +\frac{1}{d+1} + \frac{1}{d(d+1)}\)
. Use it to re-write the fraction
\(\frac{2}{15}\)
, then show that it is an identity.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
For fractions of the form
\(\frac{2}{pq}\)
, that is, fractions with a denominator that is the product of two positive integers, the following formula can also be used:
\(\frac{2}{pq}=\frac{1}{pr} + \frac{1}{qr}\)
, where
\(r = \frac{p+q}{2}\)
. Use it to re-write the fraction
\(\frac{2}{45}\)
, then show that it is an identity.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Some students may not be sure how to find a common denominator when the denominators all involve variables. Remind them that a common denominator is a multiple of all of the denominators. If needed, students can guess possible common denominators, and then test them by checking whether they're divisible by each of the denominators.
Activity Synthesis
The purpose of this discussion is for students to share how they proved the given formulas are identities. Ask 2–4 previously identified groups who used different methods to share how they each showed that the formula is an identity, recording their strategies for all to see.


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