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: Extend the properties of exponents to rational exponents.

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

Represent complex numbers and their operations on the complex plane.

Use complex numbers in polynomial identities and equations.

Use properties of rational and irrational numbers.

Perform arithmetic operations with complex numbers.

Represent and model with vector quantities.

Perform operations on vectors.

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

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

Reason quantitatively and use units to solve problems.

Perform operations on matrices and use matrices in applications.

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

Compute fluently with multi-digit numbers and find common factors and multiples.
problem_activity: Activity
Students generalize their observations and explanation from the previous activity to prove the medians of a triangle always coincide. It’s okay if groups don’t include all necessary details in the proof. Anything likely to be missed will be discussed in the activity synthesis.
Launch
Arrange students in groups of 3–4. Give students 2 minutes of work time to read and interpret the beginning of the problem, monitoring for groups that draw a diagram. Then, pull the class back together briefly, inviting a group to share its diagram.
Representing, Conversing: MLR7 Compare and Connect.
Use this routine to prepare students for the whole-class discussion about the proof that the medians of a triangle always coincide. After groups prove that the medians of triangle intersect at a single point, invite them to create a visual display of their work. Then ask students to quietly circulate and observe at least two other visual displays in the room. Give students quiet think time to consider what is the same and what is different about their proofs. Next, ask students to find a partner to discuss what they noticed. Listen for and amplify the language students use to justify why the proof applies to any triangle on the coordinate plane.
Design Principle(s): Cultivate conversation
Engagement: Develop Effort and Persistence.
Provide reminders or checklists that focus on increasing the length of on-task orientation in the face of distractions. For example, provide a checklist of the responsibilities of each group member with pre-assigned sides and medians.
Supports accessibility for: Attention; Social-emotional skills
Student Facing
The goal is to prove that the medians of any triangle intersect at a point. Suppose the vertices of a triangle are
\((0,0), (w,0),\)
and
\((a,b)\)
.
Each student in the group should choose 1 side of the triangle. If your group has 4 people, 2 can work together. Write an expression for the midpoint of the side you chose.
Each student in the group should choose a median. Write an expression for the point that partitions each median in a
\(2:1\)
ratio from the vertex to the midpoint of the opposite side.
Compare the coordinates of the point you found to those of your groupmates. What do you notice?
Explain how these steps prove that the 3 medians of any triangle intersect at a single point.
Student Response
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Anticipated Misconceptions
A diagram is a helpful tool.
Suggest that students share their results with their groups after each step.
Make sure students understand that the
\(2:1\)
ratio applies to the segment that
begins
with the vertex and
ends
with the midpoint of the opposite side of the triangle.
If students struggle to find the point that partitions the median in a
\(2:1\)
ratio, remind them of the notation they used in earlier activities:
\(\frac13A + \frac23B\)
. How does that apply to this problem?
Activity Synthesis
The goal of the discussion is to ensure students have incorporated all the necessary elements in their proof. Here are some questions for discussion:
“What can we conclude if we know that a single point is on 3 different lines?” (It means either the lines coincide, or the lines intersect at that point.)
“How do we know the medians don’t coincide?” (The midpoints are all different because the sides of the triangles don’t coincide.)
“Why did we test a
\(2:1\)
ratio?” (That’s the ratio we measured for the triangle in the warm-up, so we thought it might apply to all triangles.)
“Does this proof apply to a triangle that isn’t positioned like the one in the diagram?” (Yes. We can use a sequence of rigid motions to take any triangle to an image in which a vertex is at
\((0,0)\)
and one side coincides with the
\(x\)
-axis.)


correct_option_index: 10