few_shots: Example 1:
Problem/activity:
Task
Materials
One 10-frame per pair of students (see example below)
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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.
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options: Summarize and describe distributions.

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

Develop understanding of statistical variability.

Use the rules of probability to compute probabilities of compound events in a uniform probability model.

Interpret linear models.

Investigate patterns of association in bivariate data.

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 a single count or measurement variable.

Use random sampling to draw inferences about a population.

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

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

Calculate expected values and use them to solve problems.

Use probability to evaluate outcomes of decisions.
problem_activity: Activity
In this activity, students draw a bar graph and histogram, then describe the distributions shown on each display. Although the two visual displays may appear similar at first glance, there are important distinctions between the representations. Students notice differences in how we might characterize distributions in bar graphs and those in histograms, including how we describe typical values or categories. Along the way, students consolidate their understanding about categorical and numerical data.
Launch
Students will need the data on their travel methods and times, collected at the beginning of the unit. Distribute or display the data collected for these questions from the survey given earlier in the unit. Alternatively, complete the tables in the blackline master ahead of time.
Arrange students in groups of 2. Give one copy of the blackline master to each group of students. Display the data from the prior survey or the completed frequency tables for all to see or give a copy to each group of 2 students. Give students 5–6 minutes to complete the activity. Ask one partner to create a bar graph to represent the data on the class’s travel methods and the other to create a histogram to represent the data on travel times, and then answer the questions together.
Student Facing
Your teacher will provide you with some data that your class collected the other day.
Use the data to draw a histogram that shows your class’s travel times.
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Describe the distribution of travel times. Comment on the center and spread of the data, as well as the shape and features.
Use the data on methods of travel to draw a bar graph. Include labels for the horizontal axis.
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Describe what you learned about your class’s methods of transportation to school. Comment on any patterns you noticed.
Compare the histogram and the bar graph that you drew. How are they the same? How are they different?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Use one of these suggestions (or make up your own). Research data to create a histogram. Then, describe the distribution.
Heights of 30 athletes from multiple sports
Heights of 30 athletes from the same sport
High temperatures for each day of the last month in a city you would like to visit
Prices for all the menu items at a local restaurant
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
The purpose of the discussion is for students to recognize the differences between histograms and bar graphs.
Select one student to show a completed histogram and another to show a completed bar graph. Then, solicit several observations about how the two graphical displays compare. Ask questions such as:
“How are the bar graphs and histograms alike? How are they different?”
“Can we use a bar graph to display the data on travel times? Why or why not?”
“Can we use a histogram to display the data on methods of travel? Why or why not?”
Next, select a few other students to share their descriptions of the distributions shown on each type of display. Then, ask questions such as:
“How are your descriptions of the distribution for travel methods different than those for travel times?”
“Can you talk about the shape of a distribution shown on a bar graph? Can you talk about its center and spread? Why or why not?”
Students should recognize that only the distribution of numerical data can be described in terms of shape, center, or spread. We cannot analyze these features for a distribution of a categorical data on a bar graph because a bar graph does
not
use a number line. This means the bars can be drawn anywhere, in any order, and with any kind of spacing, so shape, center, and spread would have no meaning.
Action and Expression: Internalize Executive Functions.
Provide students with a graphic organizer to record differences between histograms and bar graphs.  Consider creating a shared display and keeping it visible throughout the unit.
Supports accessibility for: Language; Organization
Representing, Conversing: MLR2 Collect and Display.
As students discuss the similarities and differences between histograms and bar graphs, record and display phrases you hear in a chart, such as a Venn diagram. Use this as a reference when highlighting or revoicing academic vocabulary from this unit (frequency, bins, numerical data, categorical data). This will help students connect unique and shared characteristics of histograms and bar graphs while reinforcing mathematical language.
Design Principle(s): Maximize meta-awareness


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