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.


Mathematical concept/skill options: 
A. Write and interpret numerical expressions.
B. Generate and analyze patterns.
C. Analyze patterns and relationships.
D. Use properties of operations to generate equivalent expressions.
E. Use the four operations with whole numbers to solve problems.
F. Work with radicals and integer exponents.
G. Solve systems of equations.
H. Represent and solve problems involving multiplication and division.
I. Perform arithmetic operations on polynomials.
J. Understand the relationship between zeros and factors of polynomials.
K. Understand properties of multiplication and the relationship between multiplication and division.
L. Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
M. Solve problems involving the four operations, and identify and explain patterns in arithmetic.
N. Use polynomial identities to solve problems.
O. Apply and extend previous understandings of arithmetic to algebraic expressions.
P. Rewrite rational expressions.
Q. Interpret the structure of expressions.
R. Add and subtract within 20.
S. Analyze and solve linear equations and pairs of simultaneous linear equations.
T. Represent and analyze quantitative relationships between dependent and independent variables.
U. Solve equations and inequalities in one variable.
V. Create equations that describe numbers or relationships
W. Reason about and solve one-variable equations and inequalities.
X. Represent and solve equations and inequalities graphically.
Y. Understand solving equations as a process of reasoning and explain the reasoning.
Z. Understand and apply properties of operations and the relationship between addition and subtraction.
AA. Gain familiarity with factors and multiples.
AB. Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
AC. Work with addition and subtraction equations.
AD. Multiply and divide within 100.
AE. Work with equal groups of objects to gain foundations for multiplication.
AF. Represent and solve problems involving addition and subtraction.
AG. Write expressions in equivalent forms to solve problems.
AH. Understand the connections between proportional relationships, lines, and linear equations.


Thought: This task teaches students to understand addition as putting together counters, or adding to already placed counters.
Answer: AB

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} $

Mathematical concept/skill options: 
A. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
B. Extend understanding of fraction equivalence and ordering.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
C. Use equivalent fractions as a strategy to add and subtract fractions.
D. Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
E. Understand decimal notation for fractions, and compare decimal fractions.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
F. Develop Understanding of fractions as numbers.
Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.


Thought: This problem involves understanding fractions as numbers by comparing their quantities.
Answer: F

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?


Mathematical concept/skill options: 
A. Use place value understanding and properties of operations to perform multi-digit arithmetic. (A range of algorithms may be used.)
B. Understand place value.
C. Understand the place value system.
D. Extend the counting sequence.
E. Work with numbers 11–19 to gain foundations for place value.
F. Use place value understanding and properties of operations to perform multi-digit arithmetic.
Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.
G. Perform operations with multi-digit whole numbers and with decimals to hundredths.
H. Generalize place value understanding for multi-digit whole numbers.
Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.
I. Use place value understanding and properties of operations to add and subtract.


Thought: This set of problems asks students to perform multiplication, which is an operation, on multi-digit whole numbers.
Answer: G

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: im_task_001477

K

im_task_000157

3

fl_problem_000674

5
options: Calculate expected values and use them to solve problems.

Understand independence and conditional probability and use them to interpret data.

Draw informal comparative inferences about two populations.

Understand and evaluate random processes underlying statistical experiments.

Summarize, represent, and interpret data on two categorical and quantitative variables.

Summarize, represent, and interpret data on a single count or measurement variable.

Make inferences and justify conclusions from sample surveys, experiments, and observational studies.

Use random sampling to draw inferences about a population.

Investigate patterns of association in bivariate data.

Interpret linear models.

Use probability to evaluate outcomes of decisions.

Summarize and describe distributions.

Develop understanding of statistical variability.

Investigate chance processes and develop, use, and evaluate probability models.

Use the rules of probability to compute probabilities of compound events in a uniform probability model.
problem_activity: Activity
Following the review from the previous activity, students are asked to use these calculations to compare two groups more formally. Students are shown one quantifiable method of determining whether the two groups are relatively close or relatively very different in the discussion following the activity involving describing the difference of the measures of center as a multiple of the variability (MP2). The important idea for students to grasp from this activity is that the measures of center and measures of variability of the groups work together to give an idea of how similar or different the groups are.
As students work to compare heights of the two families, monitor for students who:
Create dot plots of the data and look at the overlap visually.
Compute measures of center to compare the groups numerically.
Also compute measures of variability to compare the measures of center for each group accounting for the spread of values.
Launch
Keep students in groups of 2.
Introduce students to the idea that both the measures of center as well as the measures of variability are important when comparing data sets by asking students about these contexts.
“Two groups of adults have mean weights that are different by 10 pounds. Are the two groups very different in weight?”
“Two groups of 8 year olds have weights that are different by 10 pounds. Are the two groups very different in weight?”
“Two groups of birds have weights that are different by 3.2 ounces (0.2 pounds). Are the two groups very different in weight?”
Without knowing more about the exact numbers, it seems that the adults might not be that different. A typical adult might weigh 180 pounds and, depending on diet and activity, even one person’s weight might go up or down by 10 pounds in a few weeks. It might be hard to tell these two groups apart even with the 10 pound difference in means.
On the other hand, 8 year olds typically weigh about 50 pounds, so a 10 pound difference can mean a lot more. A group with an average weight of 50 pounds is 25% heavier than a group with an average weight of 40 pounds.
For the birds, it might be hard to say how different the groups are unless we have more information. If the two groups contain the same type of bird (e.g., parrots), we might expect the weights to be fairly consistent, so a 3.2 ounce difference could be a lot. If the two groups each contain a variety of types (from hummingbirds up to emus), the 3.2 ounce difference might not be very much.
Based on these examples, it helps to know not only the means (or medians), but also the variability of the groups. We saw this in the previous activity with the overlap in the dot plots.
For the work in this activity, tell students to try to give some values to help back up their comparison of the two groups.
Allow students 1 minute of quiet think time to examine ways of approaching the problem, followed by partner work time and a whole-class discussion.
Representation: Internalize Comprehension.
Activate or supply background knowledge of measures of centers. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
Conversing: MLR2 Collect and Display.
Use this routine to capture the language students use as they compare the heights of these two families. Circulate and listen to student talk during small-group and whole-class discussion. Record the words and phrases students use on a display for all to see. Invite students to borrow from—or add more language to—the display throughout the remainder of the lesson. This will help students read and use mathematical language during their paired and whole-group discussions.
Design Principle(s): Optimize output (for comparison); Maximize meta-awareness
Student Facing
Compare the heights of these two families. Explain or show your reasoning.
The heights (in inches) of Noah’s family members: 28, 39, 41, 52, 63, 66, 71
The heights (in inches) of Jada’s family members: 49, 60, 68, 70, 71, 73, 77
Student Response
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Student Facing
Are you ready for more?
If Jada's family adopts newborn twins who are each 18 inches tall, does this change your thinking? Explain your reasoning.
Student Response
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Activity Synthesis
The purpose of this discussion is for students to begin to be more formal in their comparison of data from different groups. In particular, a general rule is established that will be used in this unit.
Select students to share their approaches in the sequence outlined in the Activity Narrative.
The heights of Noah’s and Jada’s families overlap more than the heights of the gymnastics and volleyball teams. The difference in means for the two families is about the same as between the two teams.
The difference between the mean heights of the volleyball and gymnastics teams is
\(78.17 - 62.75\)
, or 15.42 inches.
The difference between the mean heights of Jada’s and Noah’s families is
\(66.86 - 51.43\)
, or 15.43 inches. The variability in heights for the families is greater than the variability in heights for the teams.
Explain: The difference between the means is not enough information to know whether or not the data sets are very different. One way to express the amount of overlap is to divide the difference in means by the (larger) mean absolute deviation.
Demonstrate for students how to do this calculation for the volleyball and gymnastics teams:
The difference in means is more than 6 times the measure of variability, because
\(15.42 \div 2.46 \approx 6.3\)
.
Leave the calculation for the two teams displayed. Ask students to do the calculation for Jada’s and Noah’s families.
The difference in means is a little more than 1 time the measure of variability, because
\(15.43 \div 13.22 \approx 1.2\)
.
As a general rule, we will consider it a large difference between the data sets if the difference in means is
more than twice
the mean absolute deviation. If the mean absolute deviation is different for each group, use the larger one for this calculation.
For students who ask why twice the MAD is used rather than some other value, defer the question for later in the unit. A later lesson titled Do They Carry More? will address this question in more detail.


correct_option_index: 2