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

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fl_problem_000674

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options: Graph points on the coordinate plane to solve real-world and mathematical problems.

Analyze, compare, create, and compose shapes.

Find arc lengths and areas of sectors of circles.

Explain volume formulas and use them to solve problems.

Apply trigonometry to general triangles.

Use coordinates to prove simple geometric theorems algebraically.

Visualize relationships between two-dimensional and three-dimensional objects.

Translate between the geometric description and the equation for a conic section.

Understand congruence in terms of rigid motions.

Prove geometric theorems.

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

Reason with shapes and their attributes.

Apply geometric concepts in modeling situations.

Understand and apply the Pythagorean Theorem.

Make geometric constructions.

Understand and apply theorems about circles.

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

Understand similarity in terms of similarity transformations.

Define trigonometric ratios and solve problems involving right triangles.

Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

Experiment with transformations in the plane.

Draw, construct, and describe geometrical figures and describe the relationships between them.

Understand congruence and similarity using physical models, transparencies, or geometry software.

Solve real-world and mathematical problems involving area, surface area, and volume.

Prove theorems involving similarity.

Classify two-dimensional figures into categories based on their properties.
problem_activity: Activity
The goal of this activity is for students to see the usefulness of the triangle congruence theorems in proving other results. In this proof, students are expected to draw an auxiliary line, find two congruent triangles, determine that they are congruent by the Side-Side-Side Triangle Congruence Theorem, and then use the fact that corresponding parts of congruent triangles are congruent to finish the proof.
Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Arrange students in groups of 2. Remind students of the tips for writing proofs using triangle congruence theorems that are on display.
Representing, Conversing: MLR7 Compare and Connect.
Use this routine to help students develop the mathematical language of geometric proofs. After partners revise their drafts into a clear proof that angle
\(B\)
is congruent to angle
\(D\)
, invite them to create a visual display of their proof. Then ask partners to quietly circulate and read 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 partners to discuss what they noticed. Listen for students who reference the proof from a previous lesson that opposite sides of a parallelogram are congruent. Also, listen for and amplify the language students use to justify why triangle
\(ABC\)
is congruent to triangle
\(CDA\)
and angle
\(B\)
is congruent to angle
\(D\)
.
Design Principle(s): Cultivate conversation
Action and Expression: Provide Access for Physical Action.
Provide access to tools and assistive technologies such as dynamic geometry software. Some students may benefit from a checklist or list of steps to be able to use the software.
Supports accessibility for: Organization; Conceptual processing; Attention
Student Facing
Quadrilateral
\(ABCD\)
is a parallelogram. By definition, that means that segment
\(AB\)
is parallel to segment
\(CD\)
, and segment
\(BC\)
is parallel to segment
\(AD\)
.
Prove that angle
\(B\)
is congruent to angle
\(D\)
.
Work on your own to make a diagram and write a rough draft of a proof.
With your partner, discuss each other’s drafts.
What do you notice your partner understands about the problem?
What revision would help them move forward?
Work together to revise your drafts into a clear proof that everyone in your class could follow and agree with.
Student Response
Teachers with a valid work email address can
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for free access to Student Response.
Anticipated Misconceptions
If students struggle longer than is productive, direct them to their reference charts. What do they know about parallelograms? (Opposite sides are congruent.)
Activity Synthesis
Select partners who used the same approach and whose work is at similar levels of clarity to work together in groups of 4.
Instruct students to read the other group’s proof and decide if they agree with it, and if it could be improved to convince a skeptic. Invite each partnership to say one thing they noticed and liked about the proof, and one thing a skeptic might wonder about the proof.
If needed, brainstorm some things skeptics might be wondering about, such as:
When you did that rotation, how did you know these points had to line up?
How did you know those would be the same line?
How did you know those lines could only intersect there?
How did you know the Side-Side-Side Triangle Congruence Theorem would apply to these triangles? Did you list all three reasons?
How did you know those sides were congruent?
How does knowing the triangles are congruent help you prove the angles are congruent?
Did you use any information that wasn’t given or that you didn’t have a reason for?
Did what you proved match the problem?


correct_option_index: 9