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 to multiply a fraction or whole number by a fraction.

Answer: 
No. The expressions shown in this problem do not involve multiplying fractions with other fractions, nor whole numbers with fractions.

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_neg_0

7

fl_problem_001894_pos_0

HS_0
standard_description: Apply properties of operations as strategies to add and subtract. (Students need not use formal terms for these properties.) Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)
problem_activity: Narrative
This activity prompts students to use number lines to illustrate the decomposition of a fraction into sums of other fractions, reinforcing their work from an earlier lesson. Along the way, students recognize that one way to decompose a fraction greater than 1 is to write it as a sum of a whole number and a fraction less than 1. This insight prepares students to interpret and write mixed numbers in later activities.
Action and Expression: Develop Expression and Communication
. Provide access to colored pencils. Invite students to trace or draw each “jump” with a different color. Then invite students to circle or write out each addend in the corresponding color.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing, Organization
Launch
Display or draw this number line:
[IMG]
“What number does the point describe?” (8)
“What do you think the ‘jumps’ represent?” (The numbers that are put together to go from 0 to 8)
“What equations can we write to represent the combination of jumps?” (
\(6 + 2 =8\)
or
\(2 + 6 = 8\)
)
“Let’s look at the jumps on some other number lines and see what they may represent.”
Activity
Groups of 2
“Work independently on the activity for a few minutes. Afterwards, share your responses with your partner.”
5–7 minutes: independent work time
2 minutes: partner discussion
Student Facing
On each number line, draw two “jumps” to show how to use sixths to make a sum of
\(\frac{8}{6}\)
. Then, write an equation to represent each combination of jumps.
[IMG]
[IMG]
Noah draws the following diagram and writes:
\(\frac{8}{6} = \frac{6}{6} + \frac{2}{6}\)
and
\(\frac{8}{6} = 1 + \frac{2}{6}\)
. Which equation is correct? Explain your reasoning.
[IMG]
On each number line, draw “jumps” to show how to use thirds to make a sum of
\(\frac{7}{3}\)
. Then, write an equation to represent each combination of jumps.
[IMG]
[IMG]
Write
\(\frac{7}{3}\)
as a sum of a whole number and a fraction.
Student Response
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Advancing Student Thinking
Students may not make a connection between the fractions in the problems and the number lines because the latter show no fractional labels. Consider asking students to label the number line to show fractions or asking, “Where do you think
\(\frac{1}{6}\)
would be labeled on this number line?”
Activity Synthesis
Invite students to share their equations. Record them for all to see.
Focus the discussion on part b: writing
\(\frac{7}{3}\)
as a sum of a whole number and a fraction. Students are likely to write
\(1 + \frac{4}{3}\)
and
\(2 + \frac{1}{3}\)
.
Explain that
\(2 + \frac{1}{3}\)
can be written as
\(2\frac{1}{3}\)
, which we call a
mixed number
. This number is equivalent to
\(\frac{7}{3}\)
.
“Why might it be called a mixed number?” (It is a mix of a whole number and a fraction.)


aligns: no