few_shots: 
Example 1:
Problem/activity:
You’re mixing ingredients for cookies. The recipe says to combine 6 tablespoons, or ¾ stick, of butter with 1 cup of sugar. You accidentally mix in a whole stick of butter (8 tablespoons) with the cup of sugar. How can you fix this?


Concept/skill: 
Recognize and represent proportional relationships between quantities.

Answer: 
No. Usually problems that teach recognizing and representing proportional relationships between quantities involve deciding whether two quantities are in a proportional relationship or identifying the constant of proportionality, and then representing proportional relationships by equations or explaining proportional relationship using coordinate graphs.

Example 2:
Problem/activity:
For each pair of numbers, determine if the values are equivalent. Explain your reasoning for each one.
a.
$$-0.25$$
and
$$\frac{1}{-4}$$
b.
$$-\left ( \frac{1}{4} \right )$$
and
$$\frac{-1}{-4}$$
c.
$$\frac{-1}{4}$$
and
$$-\frac{1}{4}$$


Concept/skill: 
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

Answer: 
No. The expressions shown in this problem do not involve multiplying fractions with other fractions, nor whole numbers with fractions.

Example 3:
Problem/activity:
Problem 1
The table below contains single equations in a single variable, single equations in two variables, and pairs of simultaneous equations in two variables. Check the appropriate box to show whether there is no solution, exactly 1 solution, or if there is an infinite number of solutions.
{"table": [{"_attributes": {"align": "center", "class": ["table", "table-bordered"], "style": "table-layout:fixed; width:70%"}, "tbody": [{"tr": [{"td": [{"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "No solution"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Exactly 1 solution"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Infinite number of solutions"}]}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9=4x+7}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${y=3y+5}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${y={x\\over2}}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${3x+6y-9=0}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${x=y}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${1.25x=1.25y}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9.9=6x+8y}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9x=2.5y-8.8}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${11x-2y=1.5}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${11x-2y=2.5}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {}]}]}]}]}
Problem 2
Kim has a small container and a large container as shown.
[IMG]
It takes 16 of the small containers to fill the large container. Three small containers leave 1.95 gallons of space in the big container.
What is the size of each of the two containers?


Concept/skill: 
Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

Answer: 
No. Completing neither of these problems requires replacing one equation by the sum of that equation and a multiple of the other equation.

Now, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_000972_neg_0

6

fl_problem_001097_neg_0

7

fl_problem_001392_neg_0

8
standard_description: Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.
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
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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


aligns: no