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.


Your response:
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.


Your response:
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.


Your response:
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: Multiply and divide within 100.

Gain familiarity with factors and multiples.

Work with radicals and integer exponents.

Reason about and solve one-variable equations and inequalities.

Rewrite rational expressions.

Represent and solve problems involving multiplication and division.

Understand and apply properties of operations and the relationship between addition and subtraction.

Understand properties of multiplication and the relationship between multiplication and division.

Represent and analyze quantitative relationships between dependent and independent variables.

Solve systems of equations.

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Solve equations and inequalities in one variable.

Perform arithmetic operations on polynomials.

Write and interpret numerical expressions.

Generate and analyze patterns.

Use the four operations with whole numbers to solve problems.

Understand the connections between proportional relationships, lines, and linear equations.

Represent and solve equations and inequalities graphically.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Understand the relationship between zeros and factors of polynomials.

Analyze and solve linear equations and pairs of simultaneous linear equations.

Add and subtract within 20.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Work with equal groups of objects to gain foundations for multiplication.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Work with addition and subtraction equations.

Analyze patterns and relationships.

Write expressions in equivalent forms to solve problems.

Use polynomial identities to solve problems.

Represent and solve problems involving addition and subtraction.

Use properties of operations to generate equivalent expressions.

Understand solving equations as a process of reasoning and explain the reasoning.

Create equations that describe numbers or relationships

Interpret the structure of expressions.
problem_activity: Activity
In earlier activities, students wrote expressions to describe visual patterns by first creating tables and reasoning repeatedly about how the pattern changes at incrementally greater step numbers. Here they write an equation to describe a pattern without creating a table of values. They look more directly at the structure of the visual pattern and its connections to the parts of the equations, and then reason in the other direction: given an equation, they generate a visual pattern. Students also begin to frame the relationship between the quantities as a function with inputs and outputs leading to a definition of
quadratic function
.
Some students are likely to view the
\(n\)
th step of the given pattern as an
\(n\)
-by-
\(n\)
square with four small squares added at the corners, leading to the expression
\(n^2 + 4\)
for the total number of small squares. Others may view this as an
\(n+2\)
by
\(n+2\)
square with 4 rectangles removed, each consisting of
\(n\)
squares. The latter leads to the (equivalent) expression
\((n+2)^2 - 4n\)
. In the next activity, students will look closely at how equivalent expressions arise when analyzing geometric patterns, but if equivalent expressions come up in this activity, consider inviting students to share them.
This prompt gives students opportunities to see and make use of structure (MP7) when writing an equation from a visual pattern. The specific structure they might notice is an inner square with a side length equal to the step number along with a small square on each corner. Later, generating a pattern given an equation requires them to reason abstractly and concretely (MP2).
Launch
Give students a moment to observe the pattern from the activity and ask them what they notice and what they wonder. Then, ask students to sketch the next step in the pattern and share their sketch with a partner.
Representation: Internalize Comprehension.
Activate or supply background knowledge. Provide students with access to a simple two-column graphic organizer in which one column is for what they know and one column is for what they wonder. Allow students to organize their thinking this way for each step of the visual pattern.
Supports accessibility for: Visual-spatial processing; Organization
Student Facing
[IMG]
If the pattern continues, what will we see in Step 5 and Step 18?
Sketch or describe the figure in each of these steps.
How many small squares are in each of these steps? Explain how you know.
Write an equation to represent the relationship between the step number
\(n\)
and the number of squares
\(y\)
. Be prepared to explain how each part of your equation relates to the pattern. (If you get stuck, try making a table.)
Sketch the first 3 steps of a pattern that can be represented by the equation
\(y = n^2 - 1\)
.
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?
For the original step pattern in the statement, write an equation to represent the relationship between the step number
\(n\)
and the perimeter,
\(P\)
.
For the step pattern you created in part 3 of the activity, write an equation to represent the relationship between the step number
\(n\)
and the perimeter,
\(P\)
.
Are these linear functions?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Some students may wonder how to draw a pattern given the equation
\(y=n^2-1\)
.  Show them the warm-up problems where they subtracted 1 to remove the small square in Figure B and added 1 when there was an extra small square in Figure C.  Prompt them to describe subtracting 1 as removing one small square from each step.  Some students may find it easier to start drawing step 2 or step 3.  They can work backwards to draw step 1 which would have 0 squares since
\(1^2-1=0\)
.  Emphasize that making a table can help them figure out exactly how many small squares are needed in each step.
Activity Synthesis
Make sure students see the connection between the equation
\(y=n^2 +4\)
and composition of the squares in the pattern: that regardless of what
\(n\)
is, the figure at Step
\(n\)
is composed of a square that is
\(n\)
by
\(n\)
, plus 4 small squares at each corner.
Next, help students relate the work so far to the idea of functions. Discuss with students:
“In each pattern we’ve seen, is the relationship between the step number and the number of squares a function? How do you know?” (Yes. The step number is the input, and the number of squares is the output. For every step number, there is a particular number of squares.)
“How can the relationship be expressed using function notation?” (If the function
\(f\)
gives the number of squares at step number
\(n\)
, we can define it as
\(f(n) = n^2+4\)
.)
Introduce
quadratic function
as a function that is defined by a quadratic expression. Like other functions, it can be represented with an equation, a table of values, a graph, and a description.
Conversing: MLR2 Collect and Display.
Prepare students for the discussion by inviting them to first explain to a partner how each part of their equation relates to the pattern. Record the words, phrases, and diagrams students use to explain the equations they wrote to represent the pattern. Display the collected language for all to see, and 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 partner and whole-class discussions.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness


correct_option_index: 33