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.
shot_info: im_task_001477

K

im_task_000157

3

fl_problem_000674

5
options: Use square root and cube root symbols to represent solutions to equations of the form x^(2) = p and x^(3) = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that Ö2 is irrational.

Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.

Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3^(2) × 3^(–5) = 3^(–3) = 1/3^(3) = 1/27.

Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 × 108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger.
problem_activity: Activity
This activity is the third of three activities in which students investigate the value of
\(\sqrt{2}\)
. In the previous activity, students saw that
\(\frac75\)
is a pretty good approximation for
\(\sqrt2\)
.
\(\sqrt2\)
is a number when multiplied by itself equals 2, and
\(\frac75\)
multiplied by itself is pretty close to 2, it’s
\(\frac{49}{25}\)
. In the discussion of the previous activity, students also learned (or were reminded of) the definition of rational number: a fraction or its opposite.
In this activity, students have a chance to look for some more rational numbers that are close to
\(\sqrt2\)
. The idea is that they will look, for a while, for a fraction that can be multiplied by itself where the product is exactly 2. Of course, they won’t find one, because there is no such number. In the whole-class discussion that follows,
irrational number
is defined.
Launch
For this activity, it is best if students do not have access to a calculator with a square root button. A calculator that shows 9 decimal places will tell you that
\(\sqrt2=1.414213562\)
(which is not true) and that
\(1.414213562^2=2\)
(which is also not true). (To convince students that the calculator is lying to them, you have to make them multiply 1.414213562 by itself by hand, so better to sidestep the issue for now.) It
would
be handy for students to refer to a list of perfect squares while working on this activity, so consider posting such a list and drawing students’ attention to it.
Before they get started, remind students that a rational number is a fraction or its opposite, for example,
\(\frac98\)
. Let them know that terminating decimals are also rational, for example,
\(0.7 = \frac{7}{10}\)
.
As students work, it is possible they will focus on numbers in decimal form when searching for a rational number close to
\(\sqrt2\)
. If you see this, encourage students to remember their work in the previous activity and the example of
\(\frac75\)
, whose square,
\(\frac{49}{25}\)
, was very close to 2. Are there any other values like
\(\frac75\)
that might be even closer to
\(\sqrt2\)
?
Since students could search indefinitely for a solution to the last problem with no success, ask students to stop their work in order to leave 3–4 minutes for a whole-class discussion.
Representation: Develop Language and Symbols.
Display or provide charts with symbols and meanings. For example, display the term rational number with its definition and examples from the previous activity.
Supports accessibility for: Conceptual processing; Memory
Conversing, Reading: MLR2 Collect and Display.
As students work in pairs on the task, circulate and listen to as they discuss strategies for finding rational numbers that are close to
\(\sqrt{2}\)
. Record the words, phrases, and quantities students use on a visual display. Invite students to review the display, and ask questions to clarify the meaning of a word or phrase. For example, a phrase such as: “There is no number that is equal to
\(\sqrt{2}\)
” can be restated as “We could not find a rational number that is equal to
\(\sqrt{2}\)
." Encourage students to refer back to the visual display during whole-class discussions throughout the lesson and unit. This routine will provide feedback to students in a way that supports sense-making while simultaneously increasing meta-awareness of language.
Design Principle(s): Support sense-making; Maximize meta-awareness
Student Facing
A
rational number
is a fraction or its opposite (or any number equivalent to a fraction or its opposite).
Find some more rational numbers that are close to
\(\sqrt2\)
.
Can you find a rational number that is exactly
\(\sqrt2\)
?
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?
If you have an older calculator evaluate the expression
\(\displaystyle \left(\frac{577}{408}\right)^2\)
and it will tell you that the answer is 2, which might lead you to think that
\(\sqrt{2}=\frac{577}{408}\)
.
Explain why you might be suspicious of the calculator’s result.
Find an explanation for why
\(408^2\boldcdot 2\)
could not possibly equal
\(577^2\)
. How does this show that
\(\left(\frac{577}{408}\right)^2\)
could not equal 2?
Repeat these questions for
\(\displaystyle \left(\frac{1414213562375}{10000000000000}\right)^2\neq 2,\)
an equation that even many modern calculators and computers will get wrong.
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
Ideally, students were given time to look for a rational number that is a solution to
\(x^2=2\)
. Select students with particularly close values to share their number with the class and what strategy they used to find it. Applaud students for their perseverance, but confess that no such number exists because it is an
irrational number
.
Display a number line for all to see such as the one shown here. Tell students that an
irrational number
is a number that is not rational. That is, it is a number that is not a fraction or its opposite.
\(\sqrt2\)
is one example of an irrational number. It has a location on the number line, and its location can be narrowed down (it’s a tiny bit to the right of
\(\frac{7}{5}\)
), but
\(\sqrt2\)
cannot be found on a number line by subdividing the unit interval into
\(b\)
parts and taking
\(a\)
of them. We have to define
\(\sqrt{2}\)
in a different way, such as the side length of a square with area 2 square units.
[IMG]
Just looking for rational solutions to
\(x^2=2\)
and not finding any is not a proof that
\(\sqrt2\)
is irrational. But for the purposes of this course, students are asked to take it as a fact that
\(\sqrt2\)
is irrational. In students’ future studies, they may have opportunities to understand or write a proof that
\(\sqrt2\)
is irrational.
While the activities focus specifically on
\(\sqrt{2}\)
, there is nothing particularly special about this example. The square root of a whole number is either a whole number or irrational, so
\(\sqrt{10}\)
,
\(\sqrt{67}\)
, etc., are all irrational. Beyond grade level but worth having on hand for the sake of discussion is that rational multiples of irrational numbers are also irrational, so numbers like
\(5\sqrt{7}\)
,
\(\text -\sqrt{45}\)
, and
\(\frac{\sqrt{5}}{3}\)
are also irrational.


correct_option_index: 0