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.


Concept/skill: 
Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

Answer: 
No. Though this task involves addition, it does not involve any word problems.

Example 2:
Problem/activity:
For each pair of numbers, determine if the values are equivalent. Explain your reasoning for each one.
a.
$$-0.25$$
and
$$\frac{1}{-4}$$
b.
$$-\left ( \frac{1}{4} \right )$$
and
$$\frac{-1}{-4}$$
c.
$$\frac{-1}{4}$$
and
$$-\frac{1}{4}$$


Concept/skill: 
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

Answer: 
Yes. This problem involves understanding fractions and division to divide rational numbers, such as 1, -4, 4, and -1. This problem also involves an understanding of multiplication: if p and q are integers, then -(p/q) = (-p)/q = p/(-q).

Example 3:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Answer: 
Yes. An initial step when solving for x in this problem requires the use of the addition formula for cosine on the left side of the equation.

Now, does the following problem or activity enable students to learn the full intent of the following concept or skill?

shot_info: im_task_001477_neg_0

K

fl_problem_001097_pos_0

7

fl_problem_001894_pos_0

HS_0
standard_description: Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” (Expectations for unit rates in this grade are limited to non-complex fractions.)
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
click here to register or sign in
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?


aligns: no