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.


Your response:
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.


Your response:
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.


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

Use coordinates to prove simple geometric theorems algebraically.

Understand and apply the Pythagorean Theorem.

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

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

Understand and apply theorems about circles.

Define trigonometric ratios and solve problems involving right triangles.

Prove geometric theorems.

Find arc lengths and areas of sectors of circles.

Understand congruence in terms of rigid motions.

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

Understand similarity in terms of similarity transformations.

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

Analyze, compare, create, and compose shapes.

Explain volume formulas and use them to solve problems.

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

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

Classify two-dimensional figures into categories based on their properties.

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

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

Make geometric constructions.

Experiment with transformations in the plane.

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

Apply geometric concepts in modeling situations.

Reason with shapes and their attributes.

Prove theorems involving similarity.

Apply trigonometry to general triangles.
problem_activity: Activity
The proof that if a point
\(C\)
is the same distance from
\(A\)
as it is from
\(B\)
, then
\(C\)
must be on the perpendicular bisector of
\(AB\)
is challenging. Students may struggle to make sense of what it means to prove that a point must be on a line. This proof, like the proof of the Isosceles Triangle Theorem, requires thinking carefully about how to define an auxiliary line.
Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Arrange students in groups of 2. Use the Notice and Wonder instructional routine with this image to launch the activity.
Display the image for all to see. Ask student to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.
\(\overline{AP} \cong \overline{BP}\)
[IMG]
Things students may notice:
It’s a triangle.
Segment
\(PA\)
is congruent to segment
\(PB\)
.
Angle
\(A\)
is congruent to angle
\(B\)
(by the Isosceles Triangle Theorem).
There’s a dotted line.
Things students may wonder:
Is the dotted line the line of symmetry?
Is the dotted line the angle bisector of angle
\(APB\)
?
Is the dotted line the perpendicular bisector of segment
\(AB\)
?
The purpose of this launch is to elicit the idea that special lines through
\(P\)
when segments
\(PA\)
and
\(PB\)
are congruent have many overlapping properties, which will be useful when students draw auxiliary lines later in this activity. While students may notice and wonder many things about these images, wondering about the line through
\(P\)
is an important focus of the discussion.
If students have access to GeoGebra Geometry from Math Tools, suggest that it might be a helpful tool in this activity.
Reading, Listening, Conversing: MLR6 Three Reads.
Use a modified version of this routine to support reading comprehension of this problem. Use the first read to orient students to the situation. Ask students to describe what the situation is about (Diego, Jada, and Noah want to prove that if a point
\(C\)
is the same distance from
\(A\)
as it is from
\(B\)
, then
\(C\)
must be on the perpendicular bisector of
\(AB\)
.). Use the second read to identify geometric relationships stated by Diego, Jada, or Noah. Listen for and amplify important relationships such as “In Noah’s approach,
\(CD\)
is the angle bisector of angle
\(ACB\)
” and “In Jada’s approach,
\(D\)
is the midpoint of
\(AB\)
.” After the third read, ask students to brainstorm possible questions to help Jada and Noah prove that
\(CD\)
must be the perpendicular bisector of
\(AB\)
. This routine will help students make sense of the language in the word problem and the reasoning needed to solve the problem.
Design Principle(s): Support sense-making
Representation: Internalize Comprehension.
Provide appropriate reading accommodations and supports to ensure students access to written directions, word problems, and other text-based content.
Supports accessibility
for:
Language; Conceptual processing
Student Facing
Diego, Jada, and Noah were given the following task:
Prove that if a point
\(C\)
is the same distance from
\(A\)
as it is from
\(B\)
, then
\(C\)
must be on the perpendicular bisector of
\(AB\)
.
At first they were really stuck. Noah asked, “How do you prove a point is on a line?” Their teacher gave them the hint, “Another way to think about it is to draw a line that you know
\(C\)
is on, and prove that line has to be the perpendicular bisector.”
They each drew a line and thought about their pictures. Here are their rough drafts.
Diego’s approach: “I drew a line through
\(C\)
that was perpendicular to
\(AB\)
and through the midpoint of
\(AB\)
. That line is the perpendicular bisector of
\(AB\)
and
\(C\)
is on it, so that proves
\(C\)
is on the perpendicular bisector.”
[IMG]
Jada’s approach: “I thought the line through
\(C\)
would probably go through the midpoint of
\(AB\)
so I drew that and labeled the midpoint
\(D\)
. Triangle
\(ACB\)
is isosceles, so angles
\(A\)
and
\(B\)
are congruent, and
\(AC\)
and
\(BC\)
are congruent. And
\(AD\)
and
\(DB\)
are congruent because
\(D\)
is a midpoint. That made two congruent triangles by the Side-Angle-Side Triangle Congruence Theorem. So I know angle
\(ADC\)
and angle
\(BDC\)
are congruent, but I still don’t know if
\(DC\)
is the perpendicular bisector of
\(AB\)
.”
[IMG]
[IMG]
Noah’s approach: “In the Isosceles Triangle Theorem proof, Mai and Kiran drew an angle bisector in their isosceles triangle, so I’ll try that. I’ll draw the angle bisector of angle
\(ACB\)
. The point where the angle bisector hits
\(AB\)
will be
\(D\)
. So triangles
\(ACD\)
and
\(BCD\)
are congruent, which means
\(AD\)
and
\(BD\)
are congruent, so
\(D\)
is a midpoint and
\(CD\)
is the perpendicular bisector.”
With your partner, discuss each student’s approach.
What do you notice that this student understands about the problem?
What question would you ask them to help them move forward?
Using the ideas you heard and the ways you think each student could make their explanation better, write your own explanation for why
\(C\)
must be on the perpendicular bisector of
\(A\)
and
\(B\)
.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Elena has another approach: “I drew the line of reflection. If you reflect across
\(C\)
, then
\(A\)
and
\(B\)
will switch places, meaning
\(A'\)
coincides with
\(B\)
, and
\(B'\)
coincides with
\(A\)
.
\(C\)
will stay in its place, so the triangles will be congruent.”
1. What feedback would you give Elena?
2. Write your own explanation based on Elena‘s idea.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Remind students who struggle with their critique or edits to make use of the tips from the display, such as asking for 3 statements and 3 reasons, or looking for congruent triangles.
Activity Synthesis
Focus discussion on what students proved and the implications. Display the image from the launch of this activity, and ask what must be true about this image, based on what they just proved. (The dotted line is the perpendicular bisector of
\(AB\)
. The dotted line is the line of reflection of
\(AB\)
.)
\(\overline{AP} \cong \overline{BP}\)
[IMG]
Display this image of the construction of a perpendicular bisector and ask students how what they just proved explains why this construction works. (Both circles have radius
\(AB\)
, so the intersection points of the circles are the same distance from
\(A\)
and
\(B\)
. Therefore, both intersection points are on the perpendicular bisector of
\(AB\)
.)
[IMG]
Ask students to add part 1 of the Perpendicular Bisector Theorem to their reference charts as you add it to the class reference chart:
If a point
\(C\)
is the same distance from
\(A\)
as it is from
\(B\)
,
\(C\)
must be on the
perpendicular bisector
of
\(AB\)
. (Theorem)
\(\overline{AC} \cong \overline{BC}\)
, so
\(C\)
is on the line through midpoint
\(M\)
perpendicular to
\(\overline{AB}\)
[IMG]


correct_option_index: 7