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: Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

Compute fluently with multi-digit numbers and find common factors and multiples.

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

Reason quantitatively and use units to solve problems.

Perform arithmetic operations with complex numbers.

Apply and extend previous understandings of numbers to the system of rational numbers.

Perform operations on matrices and use matrices in applications.

Use properties of rational and irrational numbers.

Perform operations on vectors.

Know that there are numbers that are not rational, and approximate them by rational numbers.

Represent and model with vector quantities.

Represent complex numbers and their operations on the complex plane.

Use complex numbers in polynomial identities and equations.

Extend the properties of exponents to rational exponents.
problem_activity: Activity
In this activity, students begin to build their own table of trigonometric ratios (ratios of side lengths in right triangles, by angle measure of one acute angle). For now, call this the right triangle table so students remember these ratios only apply to right triangles. Building these ratios from the similar triangles they are based on helps students build connections between similar triangles and trigonometry.
Measuring several examples of triangles with the same angle measurements is essential in both the digital and paper versions of this activity. In the paper version, it provides more data points to get a more accurate ratio for the given angle. In both versions, it helps students understand that all right triangles with their angle as one of the angles will be similar and thus have the same ratios.
In the next activity and the cool-down of this lesson, students will look for patterns in the right triangle table, and also apply the ratios in the right triangle table to find unknown angle measures given a ratio of side lengths. Do not use the word trigonometry or any of the names of the ratios yet.
Digital
Launch
Provide access to devices that can run GeoGebra. Students will be working with the applet at
ggbm.at/bfxeb6cr
.
Arrange students in groups of 2-4. Invite students to create a spreadsheet to organize their data and perform calculations. Students can access a GeoGebra spreadsheet in the math tool kit or at
geogebra.org/spreadsheet
. For students needing extra support, consider providing this spreadsheet for them to use:
ggbm.at/jvvqrngb
.
Assign each group a pair of complementary angles. If students did the optional lesson choose to skip 30 and 60 or to give them to a group that needs extra practice.
Explain to students that the computer measurement is precise to at least the ten-thousandths place, but the thousandths place is sufficient for our needs.
Representation: Internalize Comprehension.
Use color coding and annotations to highlight connections between representations in a problem. For example color code the side adjacent to angle
\(A\)
, the side opposite angle
\(A\)
, and the hypotenuse.
Supports accessibility for: Visual-spatial processing.
Student Facing
Your teacher will give you some angles.
Use the applet to build 4 different right triangles for each of your angles.
Record the side lengths of each of the triangles.
Compute these 3 quotients for the acute angles in each triangle:
The length of the leg adjacent to your angle divided by the length of the hypotenuse
The length of the leg opposite from your angle divided by the length of the hypotenuse
The length of the leg opposite from your angle divided by the length of the leg adjacent to your angle
Find the mean of each type of quotient.
What do you notice? What do you wonder?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Arrange students in groups of 3-4. Give each group a copy of the blackline master to complete.
Assign each group a pair of complementary angles. If students did the optional lesson choose to skip 30 and 60 or to give them to a group that needs extra practice.
Discuss rounding with students before they begin. Ask students whether it makes sense to measure the short side of a triangle as about 5.2 cm and the hypotenuse as pretty close to 7.1 cm and then report that the ratio of the short side to the hypotenuse is exactly 0.73239437. (No, that's too many digits to report.)
Explain to students that the rule of thumb used by scientists is that if your measurements have two digits then your calculated answer should have two digits as well. For example, if your tools measure to the tenths place, say 5.2 cm, so that your measurements have one digit you’re confident in - the 5 in the one’s place, and one digit you may have had to estimate a bit - the 2 in the tenths place, then your answer should be rounded to two digits - 0.73. In the case where the number starts with a zero we count digits starting at the first non-zero digit, so in this case we will round to the hundredths place.
As students finish their measurements, have them find the mean of each column and record that on the class Right Triangle Table.
Check students’ mean data before they enter it to ensure that it is close to the actual values:
|   angle    | adjacent leg                           \(\div\)                          hypotenuse | opposite leg                           \(\div\)                          hypotenuse | opposite leg                           \(\div\)                          adjacent leg |
|------------|-------------------------------------------------------------------------------------|-------------------------------------------------------------------------------------|---------------------------------------------------------------------------------------|
| 10 degrees |                                        0.985                                        |                                        0.174                                        |                                         0.176                                         |
| 20 degrees |                                        0.940                                        |                                        0.342                                        |                                         0.364                                         |
| 30 degrees |                                        0.866                                        |                                        0.500                                        |                                         0.577                                         |
| 40 degrees |                                        0.766                                        |                                        0.643                                        |                                         0.839                                         |
| 50 degrees |                                        0.643                                        |                                        0.766                                        |                                         1.192                                         |
| 60 degrees |                                        0.500                                        |                                        0.866                                        |                                         1.732                                         |
| 70 degrees |                                        0.342                                        |                                        0.940                                        |                                         2.747                                         |
| 80 degrees |                                        0.174                                        |                                        0.985                                        |                                         5.671                                         |
This table is needed for the next activity, so it should be recorded to display at that time.
Representation: Internalize Comprehension.
Use color coding and annotations to highlight connections between representations in a problem. For example color code the side adjacent to angle
\(A\)
, the side opposite angle
\(A\)
, and the hypotenuse.
Supports accessibility for: Visual-spatial processing.
Student Facing
Your teacher will give you some angles.
Draw several right triangles using the angles you receive.
Precisely measure the side lengths of the triangles.
Complete the tables by computing 3 quotients for the acute angles in each triangle:
The length of the leg adjacent to your angle divided by the length of the hypotenuse
The length of the leg opposite from your angle divided by the length of the hypotenuse
The length of the leg opposite from your angle divided by the length of the leg adjacent to your angle
Find the mean of each column in your table.
What do you notice about your table? What do you wonder about your table?
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
Some students might struggle to identify opposite and adjacent legs. Prompt students to highlight the angle in question to accurately identify sides. If you point to “your angle” then the leg you are touching is adjacent and the leg you are not touching is opposite. The hypotenuse is always the longest side.
[IMG]
Activity Synthesis
Invite groups to share what they noticed and wondered about their data. (Each ratio is about the same for the angle. The ratio of the opposite side to the hypotenuse for angle
\(A\)
is the same as the ratio of the adjacent side to the hypotenuse for angle
\(B\)
.)
Speaking: MLR8 Discussion Supports.
Give students additional time to make sure that everyone in their group can explain what they noticed and wondered about the relationships between the quantities represented in the table. Invite groups to rehearse what they will say when they share with the whole class. Rehearsing provides students with additional opportunities to speak and clarify their thinking, and will improve the quality of explanations shared during the whole-class discussion.
Design Principle(s): Support sense-making; Cultivate conversation


correct_option_index: 3