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.


Standard options: 
A. 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.
B. For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.
C. Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).
D. Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
E. Fluently add and subtract within 5.


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

Example 2:
Problem/activity:
Task
 Arrange the fractions in order from least to greatest. Explain your answer with a picture.
 $\frac{1}{5}, \frac{1}{7}, \frac{1}{3} $
 $\frac{2}{5}, \frac{2}{7}, \frac{2}{3} $
 $\frac{5}{6}, \frac{3}{6}, \frac{1}{6} $
 $\frac{5}{12}, \frac{8}{12}, \frac{4}{12} $

Standard options: 
A. Understand a fraction as a number on the number line; represent fractions on a number line diagram.
Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
B. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
C. Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.


Thought: By arranging the fractions in this problem, students learn to compare fractions by reasoning about their size.
Answer: B

Example 3:
Problem/activity:
Problem 1
Solve using the standard algorithm. If you get stuck, use an area model and/or the partial products algorithm to help. Then assess the reasonableness of your answer.
a.
$$5,803\times46$$
b.
$$63\times2,116$$
Problem 2
Dunkin' introduced a croissant doughnut in 2014. In the first three months of its debut, they sold 7,870 dozen of them per day. How many croissant doughnuts were sold in the first three months?


Standard options: 
A. Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
B. Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
C. Fluently multiply multi-digit whole numbers using the standard algorithm.


Thought: This set of problems involves multi-digit whole numbers because the numbers in each problem are four and two digit integers, and the mathematical calculations involved in both problems focus on multiplication. The first problem explicitly asks the student to solve its subproblems using the standard algorithm.
Answer: C

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: im_task_001477

K

im_task_000157

3

fl_problem_000674

5
options: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.
problem_activity: In Class Launch
Use after Unit 2, Lesson 3
Elicit students’ prior knowledge about the solar system and what it looks like. Sketch a picture like this of the sun and a few planets with their orbital paths.
[IMG]
Using student input and making corrections if necessary, write a list of the planets in order from nearest to farthest away from the sun (Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, and Neptune). Ask students to estimate how long it takes each planet to go around the sun, and write the estimates next to each planet.
Tell students that there is a relationship between a planet’s distance from the sun and its orbital period (the time it takes to complete one revolution around the sun). In this activity, they will figure out how to describe that relationship using polynomials.
Teacher Instructions
Although students saw how to find equations to fit data in an earlier course, this is the first time they fit polynomial equations to data. If students need a hint about how to find a polynomial equation that fits the data better, it may help them to know that polynomials of higher degree can fit data points more closely. One good strategy is to start with a polynomial of degree 1 or 2 and then add more terms. For example, students could start with an equation of the form
\(y=nx^2\)
and then move to
\(y=nx^2+mx\)
and
\(y=nx^2+mx+k\)
. If the maximum amount of terms (1 greater than the degree) is reached and the polynomial is still not a good fit, then the degree can be increased.
Graphing technology can be a very helpful tool for finding models in this way, and students should be encouraged to use it. Graphing calculators can automatically find polynomials of specified forms that are the closest possible fit to a set of data. Desmos also has this feature. Calculators may report the value of
\(R^2\)
for the functions they produce.
\(R^2\)
measures how closely a function fits a set of data points. Tell students that the closer to 1
\(R^2\)
is, the better the function fits the data.
To conclude the task, invite students to share their models for the relationship between the orbital period and distance, and for the relationship between the orbital period and the square root of the distance. The square root should have a much simpler relationship to the orbital period: if we cube the square root of the distance and multiply it by some constant, we get the period. This is exactly what Kepler found. Share with students the statement of Kepler’s third law, which is: “The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit”. What this means is that the square of the orbital period is some constant times the cube of the distance. This can be expressed as
\(p^2 = n \boldcdot d^3\)
. This can be rearranged into something of the form
\(p= k (d^{1/2})^3\)
, which is the relationship that students have been investigating.
Once Kepler saw this relationship, he could divide the square of any planet’s period by the cube of its distance in order to find out what the proportionality constant is. Students can try this for themselves and verify that all planets have the same proportionality constant. They may also want to compare what happens with Jupiter’s moons—do they have the same proportionality constant as the planets? Newton’s laws of gravity later explained why this relationship between distance and period is true.
Student-Facing Statement
In the early 1600s, Johannes Kepler (1571–1630) studied the motions of the planets to find a good mathematical model for them. In 1619, he published his third law of planetary motion, which says how the orbital periods of the planets are related to their distances from the sun. In Kepler’s time, Uranus and Neptune had not been discovered, but here is the data for all 8 planets:
| planet  | distance (millions of km) | period (days) |
|---------|---------------------------|---------------|
| Mercury |           57.9            |     88.0      |
|  Venus  |           108.2           |     224.7     |
|  Earth  |           149.6           |     365.2     |
|  Mars   |           227.9           |     687.0     |
| Jupiter |           778.6           |     4,331     |
| Saturn  |          1,433.5          |    10,747     |
| Uranus  |          2,872.5          |    30,589     |
| Neptune |          4,495.1          |    59,800     |
Plot the distance (
\(x\)
) and period (
\(y\)
) of each planet, and find a polynomial model that fits the data as well as possible. You may have to experiment with both the degree of your polynomial function and the number of terms.
Make another plot that uses the square root of each distance instead, and find a polynomial model that fits those points as well as possible.
Which model do you think is best?
Jupiter has a lot of moons. Look up the periods and distances of some of them, and use that data to make a polynomial model of the relationship between period and distance for Jupiter’s moons.
Use your model to predict the period of another one of Jupiter’s moons using the radius of its orbit.
How good was the prediction? What are some possible sources of error?
Teacher Instructions
Although students saw how to find equations to fit data in an earlier course, this is the first time they fit polynomial equations to data. If students need a hint about how to find a polynomial equation that fits the data better, it may help them to know that polynomials of higher degree can fit data points more closely. One good strategy is to start with a polynomial of degree 1 or 2 and then add more terms. For example, students could start with an equation of the form
\(y=nx^2\)
and then move to
\(y=nx^2+mx\)
and
\(y=nx^2+mx+k\)
. If the maximum amount of terms (1 greater than the degree) is reached and the polynomial is still not a good fit, then the degree can be increased.
Graphing technology can be a very helpful tool for finding models in this way, and students should be encouraged to use it. Graphing calculators can automatically find polynomials of specified forms that are the closest possible fit to a set of data. Desmos also has this feature. Calculators may report the value of
\(R^2\)
for the functions they produce.
\(R^2\)
measures how closely a function fits a set of data points. Tell students that the closer to 1
\(R^2\)
is, the better the function fits the data.
To conclude the task, invite students to share their models for the relationship between the orbital period and distance, and for the relationship between the orbital period and the square root of the distance. The square root should have a much simpler relationship to the orbital period: if we cube the square root of the distance and multiply it by some constant, we get the period. This is exactly what Kepler found. Share with students the statement of Kepler’s third law, which is: “The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit”. What this means is that the square of the orbital period is some constant times the cube of the distance. This can be expressed as
\(p^2 = n \boldcdot d^3\)
. This can be rearranged into something of the form
\(p= k (d^{1/2})^3\)
, which is the relationship that students have been investigating.
Once Kepler saw this relationship, he could divide the square of any planet’s period by the cube of its distance in order to find out what the proportionality constant is. Students can try this for themselves and verify that all planets have the same proportionality constant. They may also want to compare what happens with Jupiter’s moons—do they have the same proportionality constant as the planets? Newton’s laws of gravity later explained why this relationship between distance and period is true.
Student-Facing Statement
In the early 1600s, Johannes Kepler (1571–1630) studied the motions of the planets to find a good mathematical model for them. In 1619, he published his third law of planetary motion, which says how the orbital periods of the planets are related to their distances from the sun. In Kepler’s time, Uranus and Neptune had not been discovered, but here is the data for all 8 planets:
| planet  | distance (millions of km) | period (days) |
|---------|---------------------------|---------------|
| Mercury |           57.9            |     88.0      |
|  Venus  |           108.2           |     224.7     |
|  Earth  |           149.6           |     365.2     |
|  Mars   |           227.9           |     687.0     |
| Jupiter |           778.6           |     4,331     |
| Saturn  |          1,433.5          |    10,747     |
| Uranus  |          2,872.5          |    30,589     |
| Neptune |          4,495.1          |    59,800     |
Plot the distance (
\(x\)
) and period (
\(y\)
) of each planet, and find a polynomial model that fits the data as well as possible. You may have to experiment with both the degree of your polynomial function and the number of terms.
Make another plot that uses the square root of each distance instead, and find a polynomial model that fits those points as well as possible.
Which model do you think is best?
Jupiter has a lot of moons. Here are the periods and distances of the Galilean moons, which were discovered in 1610:
|   moon   | distance (thousands of km) | period (days) |
|----------|----------------------------|---------------|
|    Io    |           421.8            |     1.77      |
|  Europa  |           671.1            |     3.55      |
| Ganymede |           1070.4           |     7.16      |
| Callisto |           1882.7           |     16.69     |
Use the data to make a polynomial model of the relationship between period and distance for Jupiter’s moons.
Another moon of Jupiter, number LXXI, was discovered in 2018. Its distance from Jupiter is 11,483 thousand km. According to your model, what should its period be?
The actual period of the moon is 252.0 days. How close was the prediction? What are some possible sources of error?


correct_option_index: 2