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: Compute fluently with multi-digit numbers and find common factors and multiples.

Represent complex numbers and their operations on the complex plane.

Apply and extend previous understandings of numbers to the system of rational numbers.

Perform operations on matrices and use matrices in applications.

Reason quantitatively and use units to solve problems.

Extend the properties of exponents to rational exponents.

Perform operations on vectors.

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

Know that there are numbers that are not rational, and approximate them by rational numbers.

Use properties of rational and irrational numbers.

Represent and model with vector quantities.

Use complex numbers in polynomial identities and equations.

Perform arithmetic operations with complex numbers.

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
problem_activity: Activity
Students first encountered irrational numbers at the start of this unit as a way to denote the side lengths of squares. They also spent time attempting to find a number of the form
\(\frac{a}{b}\)
, where
\(a\)
and
\(b\)
are integers, that is equal to
\(\sqrt{2}\)
only to have it revealed that
\(\sqrt{2}\)
is irrational. Now that students have recently done a lot of thinking about decimal representations of rational numbers and how to convert the infinitely repeating decimal representation of a number into a fraction representation, we revisit irrationals to further the point that no such fraction representation exists for these numbers. For many irrationals, long division doesn't work as a tool to calculate a decimal approximation because there are no two integers to divide. It can be said that a square of area 2 has a side length of
\(\sqrt{2}\)
or that
\(\sqrt{2}\)
is a number that, when squared, has a value of 2 (with
\(\text-\sqrt{2}\)
being the other number). All this means is that different methods to approximate the value need to be discussed.
In this activity, students will approximate the value of
\(\sqrt{2}\)
using successive approximation and discuss how they might figure out a value for
\(\pi\)
using measurements of circles. Students will also plot these values on number lines accurately to the thousandth place to reinforce the idea that irrational numbers are numbers. Therefore, irrational numbers have a place on the number line even if they cannot be written as a fraction of integers.
Launch
Calculators are okay, but students should not use the root button or the
\(\pi\)
button. Before beginning, remind students that we have previously found decimal representations for fractions, and that knowing these representations made it easier to plot numbers on a number line. Today we are going to do the same thing with irrational numbers. Ask students “Earlier, we used long division to find decimal representations of numbers. Why can't we do that today with irrational numbers?” (Irrational numbers are ones that cannot be written as fractions.)
Follow work time with a whole-class discussion.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, present one question at a time and monitor students to ensure they are making progress throughout the activity.
Supports accessibility for: Organization; Attention
Student Facing
Why is
\(\sqrt{2}\)
between 1 and 2 on the number line?
Why is
\(\sqrt{2}\)
between 1.4 and 1.5 on the number line?
How can you figure out an approximation for
\(\sqrt{2}\)
accurate to 3 decimal places?
Label all of the tick marks. Plot
\(\sqrt{2}\)
on all three number lines. Make sure to add arrows from the second to the third number lines.
[IMG]
Elena notices a beaker in science class says it has a diameter of 9 cm and measures its circumference to be 28.3 cm. What value do you get for
\(\pi\)
using these values and the equation for circumference,
\(C=2\pi r\)
?
Diego learned that one of the space shuttle fuel tanks had a diameter of 840 cm and a circumference of 2,639 cm. What value do you get for
\(\pi\)
using these values and the equation for circumference,
\(C=2\pi r\)
?
Label all of the tick marks on the number lines. Use a calculator to get a very accurate approximation of
\(\pi\)
and plot that number on all three number lines.
[IMG]
How can you explain the differences between these calculations of
\(\pi\)
?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
The purpose of this discussion is to deepen students understanding that irrational numbers are not fractions. As such, their values are approximated using different methods than for rational numbers. Begin the discussion by asking:
“How long do you think you could keep using the method in the first problem to find more digits of the decimal representation of
\(\sqrt{2}\)
?” (My calculator only shows 9 digits to the left of the decimal, so that's as far as I could go.)
“Would this method work for any root?” (Yes, I could find the two perfect squares the square of the root is between and then start approximating the value of the root from there.)
Tell students that what they did in the first problem is a strategy called “successive approximation.” It takes time, but successive approximation works for finding more and more precise approximations of irrational numbers so long as you have a clear value to check against. In the case of
\(\sqrt{2}\)
, since
\(\sqrt{2}^2=2\)
, there was a clear value to test approximations against whether they are too high or too low.
Lastly, select 2–3 students to share their response to the last part of the second problem. Make sure students understand that since measuring has limitations of accuracy, any calculation of
\(\pi\)
using measurement, such as with the beaker and the fuel tank, will have its own accuracy limited.
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion when students to share their response to the last part of the second problem. Call on students to use mathematical language to restate and/or revoice the strategy (or strategies) presented. Consider providing students time to restate what they hear to a partner, before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students' attention to any words or phrases that helped to clarify the original statement. This will provide more students with additional opportunities to speak and describe the differences between calculations of
\(\pi\)
.
Design Principle(s): Support sense-making; Maximize meta-awareness


correct_option_index: 8