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.


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

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} $

Standard options: 
A. Understand a fraction as a number on the number line; represent fractions on a number line diagram.
Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
B. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
C. Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.


Thought: By arranging the fractions in this problem, students learn to compare fractions by reasoning about their size.
Answer: B

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?


Standard options: 
A. Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
B. Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
C. Fluently multiply multi-digit whole numbers using the standard algorithm.


Thought: This set of problems involves multi-digit whole numbers because the numbers in each problem are four and two digit integers, and the mathematical calculations involved in both problems focus on multiplication. The first problem explicitly asks the student to solve its subproblems using the standard algorithm.
Answer: C

Now, assign the following problem or activity to one or more relevant standards it aligns with.
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options: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
problem_activity: Activity
This activity allows students to see how changing the target number in a guessing game changes the absolute guessing errors and changes the scatter plot of the data.
Here, students see that they are still finding the difference between the guesses and a number, that the absolute guessing errors are still all positive, and that the data points still seem to form a V above the horizontal axis. What is different is that the position of the V shape has now shifted horizontally and that the number of points forming the two pieces of the V may have changed.
The work here prepares students to later see the absolute value function in terms of finding the distance between input values and 0.
The activity is written so that students could perform the calculations and graphing individually and by hand. To make more time for reasoning and analysis, however, use of statistical and graphing technology for computation and plotting is recommended, as is splitting up the work among students.
Launch
Tell students that suppose there had been a mistake in the reported number of items in the jar, and that their job is to find out how the absolute guessing errors and the scatter plot would change once they find out the corrected number of items.
Consider giving one half of the class one value for the actual number of objects, and giving the other half of the class another value so that they could observe the general behavior of the function. (For example, give “50” to half the class and “40” to the other half.)
Students could follow the same process as earlier: calculating the absolute errors using the new "actual" number, recording them in a table, and plotting the data points on a coordinate plane. If doing so by hand, ask students to use Table B and the second coordinate plane on the handout given earlier. Keep students in groups of 2–4 so they could split up the calculations. Provide access to calculators.
Alternatively, give students the option of using technology (statistical or graphing tool) to calculate the absolute errors and to create the scatter plot. Another option is to arrange for the calculations to be split up among students and collected in a shared table or spreadsheet, and then create a class scatter plot, either by hand or using technology, displayed for all to see.
Student Facing
Earlier, you guessed the number of objects in a container and then your teacher told you the actual number.
Suppose your teacher made a mistake about the number of objects in the jar and would like to correct it. The actual number of objects in the jar is
\(\underline{\hspace{0.5in}}\)
.
Find the new absolute guessing errors based on this new information. Record the errors in Table B of the handout (or elsewhere as directed by your teacher).
Make 1–2 observations about the new set of absolute guessing errors.
Predict how the scatter plot would change given the new actual number of objects. (Would it have the same shape as in the first scatter plot? If so, what would be different about it? If not, what would it look like?)
Use technology to plot the points and test your prediction.
Can you write a rule for finding the output (absolute guessing error) given the input (a guess)?
Student Response
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Activity Synthesis
Invite students to share their observations of the new data set (or data sets, if two new "actual" numbers were given to students).
If time is limited, focus the discussion on the features of the new scatter plot, and on whether the relationship between the guesses and the absolute guessing errors form a function.
Display a completed scatter plot for all to see. Ask questions such as:
"How is this scatter plot like the first scatter plot we saw earlier?" (The points form two straight lines that seem to converge at a point, forming a V.)
"How is this scatter plot different than the first one?" (The points seem to have been shifted horizontally. Earlier, the two sets of points—each forming a line—seemed to meet at
\((47,0)\)
. Now they see to meet at
\((50,0)\)
.)
Students may also note that there were more points (or fewer points, depending on the data) that represent underestimates (or overestimates) in the new scatter plot than in the first one.
Conversing, Writing: MLR2 Collect and Display.
Before the whole-class discussion, invite students to share with their partner their observations of the new data set and compare them with what they saw in the previous activity. Listen for and collect vocabulary and phrases students use to describe similarities and differences between the scatter plots on a visual display. Include words and phrases such as “form a V,” “no negative numbers,” and “shift to the right” on the display for all to see. Remind students to borrow language from the display as needed. This will help students discuss the features of a scatter plot using appropriate mathematical language.
Design Principle(s): Support sense-making
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Invite students to talk with their partner about their observations of the new data set before writing them down. Display sentence frames to support students when they explain their strategy. For example “One thing that is the same is . . .”, “One thing that is different is . . . .”
Supports accessibility for: Language; Social-emotional skills


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