Reasoning about Equations and Tape Diagrams Part 2: IM 7.6.5
[url=https://im.kendallhunt.com/MS/teachers/2/6/5/preparation.html]“Reasoning about Equations and Tape Diagrams (Part 2)”[/url] from IM Grade 7 by [url=https://www.geogebra.org/m/gus8ffvr]Open Up Resources[/url] and Illustrative Mathematics. Licensed under the [url=https://creativecommons.org/licenses/by/4.0/]Creative Commons Attribution 4.0 license[/url].
Table of Contents
- Lesson 7.6.5
- IM 7.6.5 Lesson: Reasoning about Equations and Tape Diagrams (Part 2)
- Practice 7.6.5
- IM 7.6.5 Practice: Reasoning about Equations and Tape Diagrams (Part 2)
IM 7.6.5 Lesson: Reasoning about Equations and Tape Diagrams (Part 2)
Solve the equation mentally.
[math]x-1=5[/math]
[math]2\left(x-1\right)=10[/math]
[math]3\left(x-1\right)=15[/math]
[math]500=100\left(x-1\right)[/math]
Draw a tape diagram to represent this situation. Each of 5 gift bags contains x pencils. Tyler adds 3 more pencils to each bag. Altogether, the gift bags contain 20 pencils.
Draw a tape diagram to represent this situation. Noah drew an equilateral triangle with sides of length 5 inches. He wants to increase the length of each side by x inches so the triangle is still equilateral and has a perimeter of 20 inches.
Draw a tape diagram to represent this situation. You may need to decide what to represent with a variable. An art class charges each student $3 to attend plus a fee for supplies. Today, $20 was collected for the 5 students attending the class.
Draw a tape diagram to represent this situation. You may need to decide what to represent with a variable. Elena ran 20 miles this week, which was three times as far as Clare ran this week. Clare ran 5 more miles this week than she did last week.
Which equation represents the following situation?[br][br]Each of 5 gift bags contains [math]x[/math] pencils. Tyler adds 3 more pencils to each bag. Altogether, the gift bags contain 20 pencils.
Find the solution to the equation from your answer above. Use your diagrams above to help you reason.
What does the solution tell you about its situation?
Each situation in the previous activity is represented by one of the equations.
Which equation represents the following situation?[br][br]Noah drew an equilateral triangle with sides of length 5 inches. He wants to increase the length of each side by [math]x[/math] inches so the triangle is still equilateral and has a perimeter of 20 inches.
Find the solution to the equation from your answer above. Use your diagrams above to help you reason.
What does the solution tell you about its situation?
Which equation represents the following situation?[br][br]An art class charges each student $3 to attend plus a fee for supplies. Today, $20 was collected for the 5 students attending the class.
Find the solution to the equation from your answer above. Use your diagrams above to help you reason.
What does the solution tell you about its situation?
Which equation represents the following situation?[br][br]Elena ran 20 miles this week, which was three times as far as Clare ran this week. Clare ran 5 more miles this week than she did last week.
Find the solution to the equation from your answer above. Use your diagrams above to help you reason.
What does the solution tell you about its situation?
Han, his sister, his dad, and his grandmother step onto a crowded bus with only 3 open seats for a 42-minute ride. They decide Han’s grandmother should sit for the entire ride. Han, his sister, and his dad take turns sitting in the remaining two seats, and Han’s dad sits 1.5 times as long as both Han and his sister. How many minutes did each one spend sitting?
IM 7.6.5 Practice: Reasoning about Equations and Tape Diagrams (Part 2)
Here are some prices customers paid for different items at a farmer’s market. Find the cost for 1 pound of each item.
$5 for 4 pounds of apples
$3.50 for [math]\frac{1}{2}[/math] pound of cheese
$8.25 for [math]1\frac{1}{2}[/math] pounds of coffee beans
$6.75 for [math]\frac{3}{4}[/math] pounds of fudge
$5.50 for a [math]6\frac{1}{4}[/math] pound pumpkin
Find the products.
[math]\frac{2}{3}\cdot\left(\frac{-4}{5}\right)[/math]
[math]\left(\frac{-5}{7}\right)\cdot\left(\frac{7}{5}\right)[/math]
[math]\left(\frac{-2}{39}\right)\cdot39[/math]
[math]\left(\frac{2}{5}\right)\cdot\left(\frac{-3}{4}\right)[/math]
A family buys 6 tickets to a show. They also each spend $3 on a snack. They spend $24 on the show.
Which equation represents the story?
What does [math]x[/math] represent in the equation you picked above?
Find the solution to the equation you picked above. Explain or show your reasoning.
What does the solution tell you about its situation?
Diego has 24 ounces of juice. He pours equal amounts for each of his 3 friends, and then adds 6 more ounces for each.
Which equation represents the story?[br]
What does [math]x[/math] represent in the equation you picked above?
Find the solution to the equation you picked above. Explain or show your reasoning.
What does the solution tell you about its situation?
[size=150]Here is a diagram and its corresponding equation. [/size][br][br][img]data:image/png;base64,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[/img] [math]6\left(x+1\right)=24[/math][br][br]Find the solution to the equation and explain your reasoning.
[size=150]Below is a set of data about temperatures. The [i]range[/i] of a set of data is the distance between the lowest and highest value in the set. What is the range of these temperatures?[/size][br][br][math]9^\circ \text{C}, \text-3^\circ \text{C}, 22^\circ \text{C}, \text-5^\circ \text{C}, 11^\circ \text{C}, 15^\circ \text{C}[/math]
[size=150]A store is having a 25% off sale on all shirts. Explain or show two different ways to calculate the sale price for a shirt that normally costs $24.[/size]