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 problem or activity shown below 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: Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.
problem_activity: Activity
The goal of this activity is for students to recognize rewriting a quadratic expression from standard form to vertex form essentially entails completing the square.
Students had quite a bit of experience completing the square, mostly in the context of solving equations, in which they know to add or subtract the same number from both sides of an equation to keep the equation true. Here students are dealing with expressions and need to be careful to keep each expression equivalent to the original. If they add a number to the expression, they need to remember to subtract the same number to keep the value of the expression unchanged.
Launch
Arrange students in groups of 2. Give students quiet time to think about the first two questions and then time to share their thinking with a partner.
Pause for a class discussion after the second question and invite one or more students to demonstrate their strategy. Highlight approaches that involve completing the square. If no students took that approach, display and discuss the following:
\(\displaystyle \begin {align} &x^2 - 6x + 2\\ &x^2 - 6x + 9 - 9 + 2 \\&(x - 3)^2 - 7 \end {align}\)
Then, ask students to proceed with the rest of the activity. Provide access to devices that can run Desmos or other graphing technology.
Representation: Internalize Comprehension.
Use color and annotations to illustrate student thinking. As students share their reasoning about how to convert standard form back into vertex form, scribe their thinking on a visible display. Use annotations and labels to show the structure of the steps using the form included in the launch. For example, labeling the first step as “standard form,” labeling +9 as “add nine to complete the square,” and drawing an arrow to the -9 and labeling it as “keeping it equivalent.” Continue to add in any key phrases students contributed that illustrate the steps.
Supports accessibility for: Visual-spatial processing; Conceptual processing
Student Facing
Here are two expressions in vertex form. Rewrite each expression in standard form. Show your reasoning.
\((x+5)^2+1\)
\((x-3)^2-7\)
Think about the steps you took, and about reversing them. Try converting one or both of the expressions in standard form back into vertex form. Explain how you go about converting the expressions.
Test your strategy by rewriting
\(x^2+10x+9\)
in vertex form.
Let’s check the expression you rewrote in vertex form.
Use graphing technology to graph both
\(x^2+10x+9\)
and your new expression. Does it appear that they define the same function?
If you convert your expression in vertex form back into standard form, do you get
\(x^2+10x+9\)
?
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 may wonder why they need to both add and subtract a number from the standard form expression in order to complete the square. In previous lessons, students completed the square while solving an equation, often adding the same number to each side of the equal sign to maintain equality. Here, there is no equal sign. Emphasize that each move must generate an equivalent expression. Show a few expressions such as 5,
\(5+0\)
,
\(5-2+2\)
,
\(5+7-7\)
,
\(5+8\)
, and
\(5+8-3\)
. Ask them which ones are equal to 5. Next, ask them to write 2 more expressions that include the number 5 and also equal 5. Encourage students to notice that the sum of the numbers excluding 5 must be 0.
Activity Synthesis
Invite students to share their graph and response to the last question. Emphasize that while graphing is a quick way to check whether two expressions define the same function, it is not always reliable. If we make an algebraic error, the graph would almost certainly show it. But having two graphs that
appear to be
identical does not
prove
that two expressions are indeed equivalent. We can only be sure that two expressions define the same function by showing equivalence algebraically. For example, when students convert the expression in vertex form back to standard form, does it produce the original expression?
Make sure students understand that whatever operation is performed on an expression to complete the square, it should not change the value of the expression. Adding opposites (for example, 9 and -9, or -25 and 25), or adding and subtracting the same number, has the effect of adding 0, which keeps the original and the transformed expressions equivalent.


aligns: no