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.


Standard options: 
A. Represent addition and subtraction with objects, fingers, mental images, drawings (Drawings need not show details, but should show the mathematics in the problem. (This applies wherever drawings are mentioned in the Standards.)), sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.
B. For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.
C. Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).
D. Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
E. Fluently add and subtract within 5.


Your response:
Thought: This task involves representing addition with objects such as number cards and dice.
Answer: A

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
$$-\frac{1}{4}$$


Standard options: 
A. Solve real-world and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.)
B. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
C. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.


Your response:
Thought: 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).
Answer: B

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


Standard options: 
A. Prove the Pythagorean identity sin^(2)(θ) + cos^(2)(θ) = 1 and use it to find sin(θ), cos(θ),or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
B. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.


Your response:
Thought: 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. Solving this problem does not require use of the Pythagorean identity.
Answer: B

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: im_task_001477

K

fl_problem_001097

7

fl_problem_001894

HS_0
options: Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2).

Fluently divide multi-digit numbers using the standard algorithm.
problem_activity: Activity
In this activity, students use long division to divide whole numbers whose quotient is not a whole number. Previously, students found the quotient of
\(62 \div 5\)
using base-ten diagrams and the partial quotients method. Because the long division is a particular version of the partial quotients method, and because students have been introduced to long division, they have the tools to divide whole numbers that result in a (terminating) decimal. In this activity, students evaluate and critique the reasoning of others (MP3).
Teacher Notes for IM 6–8 Accelerated
Adjust this activity to 15 minutes. Consider skipping the third and fourth sets of questions in this activity to save time for the next activity, where students will continue practicing these skills with problems where the dividend is a decimal.
Launch
Arrange students in groups of 2. Provide access to graph paper. Give students 7–8 minutes to analyze and discuss Lin’s work with a partner and then complete the second set of questions.
Pause to discuss students’ analyses and at least one of the division problems. Students should understand that, up until reaching the decimal point, long division works the same for
\(62 \div 5 = 12.4\)
as it does for
\(657 \div 3 = 219\)
. In
\(62 \div 5\)
, however, there is a remainder of 2 ones, and we need to convert to the next smaller place value (tenths), change the 2 ones into 20 tenths, and then divide these into 5 equal groups of 4 tenths.
Prepare students to do the last set of questions by setting up the long division of
\(5 \div 4\)
for all to see. Discuss the placement of the decimal point. Reiterate that we can write extra zeros at the end of the dividend, following the decimal point, so that remainders can be worked with. Have students work through this problem and the others. Follow with a whole-class discussion.
Representation: Access for Perception.
Display and read Lin’s method for calculating the quotient of
\(62 \div 5\)
aloud. Students who both listen to and read the information will benefit from extra processing time.
Supports accessibility for: Language; Conceptual processing
Student Facing
Here is how Lin calculated
\(62 \div 5\)
.
[IMG]
Discuss with your partner:
Lin put a 0 after the remainder of 2. Why? Why does this 0 not change the value of the quotient?
Lin subtracted 5 groups of 4 from 20. What value does the 4 in the quotient represent?
What value did Lin find for
\(62 \div 5\)
?
Use long division to find the value of each expression. Then pause so your teacher can review your work.
\(126 \div 8\)
\(90 \div 12\)
Use long division to show that:
\(5 \div 4\)
, or
\(\frac 54\)
, is 1.25.
\(4 \div 5\)
, or
\(\frac 45\)
, is 0.8.
\(1 \div 8\)
, or
\(\frac 18\)
, is 0.125.
\(1 \div 25\)
, or
\(\frac {1}{25}\)
, is 0.04.
Noah said we cannot use long division to calculate
\(10 \div 3\)
because there will always be a remainder.
What do you think Noah meant by “there will always be a remainder”?
Do you agree with him? Explain your reasoning.
Student Response
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Anticipated Misconceptions
Students may be perplexed by the repeating decimals in the last question and think that they have made a mistake. Ask them to compare their work with a partner’s, and then clarify during discussion that some decimals do repeat. Because this work comes into focus in grade 7, the goal here is simply for students to observe that not all decimals terminate.
Activity Synthesis
Focus the whole-class discussion on the third and fourth sets of questions. Ask a few students to show their long division for all to see and to explain their steps. Some ideas to bring to uncover:
Problems like
\(1 \div 25\)
are challenging because the first step is 0: there are
zero
groups of 25 in 1. This means that we need to introduce a decimal and put a 0 to the right of the decimal. But one 0 is not enough. It is not until we add the second 0 to the right of the decimal that we can find 4 groups of 25 in 100. Because we moved two places to the right of the decimal, these 4 groups are really
\(0.04\)
, which is the quotient of 1 by 25.
Problems like
\(1 \div 3\)
are not fully treated until grade 7. At this point, we can observe that the long division process will go on and on because there is always a remainder of 1.
Some questions to ask students that highlight these points include:
When you found
\(1 \div 8\)
, what was your first step? (I put a 0 above the dividend 1 and added a decimal because I cannot take any 8’s from 1.)
When you found
\(1 \div 25\)
, what were your first few steps? (I put a 0 above the dividend 1 and added a decimal because I cannot take any 25’s from 1. I needed to add another 0 to the right of the decimal because I still cannot take any 25’s from 10.)
Why do we not write 0’s in advance to the right of the decimal? (We do not know in advance if we will need these 0’s or how many of them we might need, so we usually write them as needed.)
Another valuable link to make is to connect the values for the decimals in Problem 4 to percentages. For example, the fact that
\(\frac{4}{5} = 0.8\)
means that
\(\frac{4}{5}\)
of a quantity is 80% of that quantity. Similarly
\(\frac{1}{25}\)
of a quantity is the same as 4% of that quantity.
Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time.
Use this routine to help students improve their written responses to the question, “Noah said we cannot use long division to calculate
\(10÷3\)
because there will always be a remainder. Do you agree or disagree with this statement?” Give students time to meet with 2–3 partners to share and get feedback on their responses. Provide listeners with prompts that will help their partners strengthen their ideas and clarify their language (e.g., “Can you explain why there is always a remainder?”, “What number always remains?”, and “Could you try to explain this using a different example?”). Give students 1–2 minutes to revise their writing based on the feedback they received.​ Communicating their reasoning with a partner will help students understand situations in which long division does not work
Design Principle(s): Optimize output (for explanation); Cultivate conversation


correct_option_index: 2