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 and division and of fractions to multiply and divide rational numbers.

Answer: 
Yes. This problem involves understanding fractions and division to divide rational numbers, such as 1, -4, 4, and -1. This problem also involves an understanding of multiplication: if p and q are integers, then -(p/q) = (-p)/q = p/(-q).

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.
|                                                          | **           No solution          ** | **           Exactly 1 solution          ** | **           Infinite number of solutions          ** |
|----------------------------------------------------------|--------------------------------------|---------------------------------------------|-------------------------------------------------------|
|                       $${9=4x+7}$$                       |                                      |                                             |                                                       |
|                       $${y=3y+5}$$                       |                                      |                                             |                                                       |
|                    $${y={x\over2}}$$                     |                                      |                                             |                                                       |
|                     $${3x+6y-9=0}$$                      |                                      |                                             |                                                       |
|    $${x=y}$$                        $${1.25x=1.25y}$$    |                                      |                                             |                                                       |
| $${9.9=6x+8y}$$                        $${9x=2.5y-8.8}$$ |                                      |                                             |                                                       |
| $${11x-2y=1.5}$$                        $${11x-2y=2.5}$$ |                                      |                                             |                                                       |
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_pos_0

7

fl_problem_001392_neg_0

8
standard_description: Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
problem_activity: Activity
This task culminates in writing a formula for the area of triangles. By now students are likely to have developed the intuition that the area of a triangle is half of that of a parallelogram with the same base and height. This activity encapsulates that work in an algebraic expression.
Students first find the areas of several triangles given base and height measurements. They then generalize the numerical work to arrive at an expression for finding the area of any triangle (MP8).
If needed, remind students how they reasoned about the area of triangles in the previous lesson (i.e. by composing a parallelogram, enclosing with one or more rectangles, etc.). Encourage them to refer to their previous work and use tracing paper as needed. Students might write
\(b \boldcdot h \div 2\)
or
\(b \boldcdot h \boldcdot \frac12\)
as the expression for the area of any triangle. Any equivalent expression should be celebrated.
At the end of the activity, consider giving students a chance to reason more abstractly and deductively, i.e., to think about why the expression
\(b \boldcdot h \div 2\)
would hold true for all triangles. See Activity Synthesis for prompts and diagrams that support such reasoning.
Launch
Arrange students in groups of 2–3. Explain that they will now find the area of some triangles using what they know about base-height pairs in triangles and the relationships between triangles and parallelograms.
Give students 5–6 minutes to complete the activity and access to geometry toolkits, especially tracing paper. Ask them to find the area at least a couple of triangles independently before discussing with their partner(s).
Student Facing
For each triangle:
Identify a base and a corresponding height, and record their lengths in the table.
Find the area of the triangle and record it in the last column of the table.
[IMG]
|              triangle               | base (units) | height (units) | area (square units) |
|-------------------------------------|--------------|----------------|---------------------|
|      **          A         **       |              |                |                     |
|      **          B         **       |              |                |                     |
|      **          C         **       |              |                |                     |
|      **          D         **       |              |                |                     |
| **          any triangle         ** |    \(b\)     |     \(h\)      |                     |
In the last row, write an expression for the area of any triangle, using
\(b\)
and
\(h\)
.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Students may not be inclined to write an expression using the variables
\(b\)
and
\(h\)
and instead replace the variables with numbers of their choice. Ask them to reflect on what they did with the numbers for the first four triangles. Then, encourage them to write the same operations but using the letters
\(b\)
and
\(h\)
rather than numbers.
Activity Synthesis
Select a few students to share their expression for finding the area of any triangle. Record each expression for all to see.
To give students a chance to reason logically and deductively about their expression, ask, “Can you explain why this expression is true for
any
triangle?”
Display the following diagrams for all to see. Give students a minute to observe the diagrams. Ask them to choose one that makes sense to them and use that diagram to explain or show in writing that the expression
\(b \boldcdot h \div 2\)
works for finding the area of any triangle. (Consider giving each student an index card or a sheet of paper on which to write their reasoning so that their responses could be collected, if desired.)
[IMG]
When dealing only with the variables
\(b\)
and
\(h\)
and no numbers, students are likely to find Jada's and Lin's diagrams more intuitive to explain. Those choosing to use Elena's diagram are likely to suggest moving one of the extra triangles and joining it with the other to form a non-rectangular parallelogram with an area of
\(b \boldcdot h\)
. Expect students be less comfortable reasoning in abstract terms than in concrete terms. Prepare to support them in piecing together a logical argument using only variables.
If time permits, select students who used different diagrams to share their explanation, starting with the most commonly used diagram (most likely Jada's). Ask other students to support, refine, or disagree with their arguments. It time is limited, consider collecting students' written responses now and discussing them in an upcoming lesson.
Representation: Develop Language and Symbols.
Create a display of important terms and concepts. Invite students to suggest language or diagrams to include that will support their understanding. Include the following terms and maintain the display for reference throughout the unit: opposite vertex, base (of a triangle), height (of a triangle), and formula for finding area.
Supports accessibility for: Memory; Language
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion when students share their expressions for finding the area of any triangle. Provide students with 1–2 minutes of quiet think time to begin to consider why their expression is true for any triangle, before they continue work with a partner to complete their response. Select 1 or 2 pairs of students to share with the class, then call on students to restate their peers' reasoning. This will give more students the chance to use language to interpret and describe expressions for the area of triangles.
Design Principle(s): Support sense-making; Maximize meta-awareness


aligns: no