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.


Mathematical concept/skill options: 
A. Write and interpret numerical expressions.
B. Generate and analyze patterns.
C. Analyze patterns and relationships.
D. Use properties of operations to generate equivalent expressions.
E. Use the four operations with whole numbers to solve problems.
F. Work with radicals and integer exponents.
G. Solve systems of equations.
H. Represent and solve problems involving multiplication and division.
I. Perform arithmetic operations on polynomials.
J. Understand the relationship between zeros and factors of polynomials.
K. Understand properties of multiplication and the relationship between multiplication and division.
L. Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
M. Solve problems involving the four operations, and identify and explain patterns in arithmetic.
N. Use polynomial identities to solve problems.
O. Apply and extend previous understandings of arithmetic to algebraic expressions.
P. Rewrite rational expressions.
Q. Interpret the structure of expressions.
R. Add and subtract within 20.
S. Analyze and solve linear equations and pairs of simultaneous linear equations.
T. Represent and analyze quantitative relationships between dependent and independent variables.
U. Solve equations and inequalities in one variable.
V. Create equations that describe numbers or relationships
W. Reason about and solve one-variable equations and inequalities.
X. Represent and solve equations and inequalities graphically.
Y. Understand solving equations as a process of reasoning and explain the reasoning.
Z. Understand and apply properties of operations and the relationship between addition and subtraction.
AA. Gain familiarity with factors and multiples.
AB. Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
AC. Work with addition and subtraction equations.
AD. Multiply and divide within 100.
AE. Work with equal groups of objects to gain foundations for multiplication.
AF. Represent and solve problems involving addition and subtraction.
AG. Write expressions in equivalent forms to solve problems.
AH. Understand the connections between proportional relationships, lines, and linear equations.


Thought: This task teaches students to understand addition as putting together counters, or adding to already placed counters.
Answer: AB

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} $

Mathematical concept/skill options: 
A. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
B. Extend understanding of fraction equivalence and ordering.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
C. Use equivalent fractions as a strategy to add and subtract fractions.
D. Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
E. Understand decimal notation for fractions, and compare decimal fractions.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
F. Develop Understanding of fractions as numbers.
Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.


Thought: This problem involves understanding fractions as numbers by comparing their quantities.
Answer: F

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?


Mathematical concept/skill options: 
A. Use place value understanding and properties of operations to perform multi-digit arithmetic. (A range of algorithms may be used.)
B. Understand place value.
C. Understand the place value system.
D. Extend the counting sequence.
E. Work with numbers 11–19 to gain foundations for place value.
F. Use place value understanding and properties of operations to perform multi-digit arithmetic.
Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.
G. Perform operations with multi-digit whole numbers and with decimals to hundredths.
H. Generalize place value understanding for multi-digit whole numbers.
Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.
I. Use place value understanding and properties of operations to add and subtract.


Thought: This set of problems asks students to perform multiplication, which is an operation, on multi-digit whole numbers.
Answer: G

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: im_task_001477

K

im_task_000157

3

fl_problem_000674

5
options: Prove and apply trigonometric identities.

Construct and compare linear and exponential models and solve problems.

Build a function that models a relationship between two quantities.

Model periodic phenomena with trigonometric functions.

Interpret functions that arise in applications in terms of the context.

Use functions to model relationships between quantities.

Build new functions from existing functions.

Understand the concept of a function and use function notation.

Define, evaluate, and compare functions.

Extend the domain of trigonometric functions using the unit circle.

Interpret expressions for functions in terms of the situation they model.

Analyze functions using different representations.
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