3 Question Types
Reversibility
1. Work backwards to create an equation where the solution is x = 2. Write two more equations one
with a solution is No Solutions and another with Infinitely Many (All) Solutions.
2. Sam used the rules of solving equations to find the solution was x = -7. Write at least 2 different
problems that might possibly have been the question asked by the teacher that led Sam to the correct
answer. Prove your problems are valid responses by solving each equation.
3. Write two rational expressions whose product is 1.
4. The area of a rectangle is x2 – x – 2, find a length and width.
5. If 25 – 25 * 7 = 49, then insert parentheses to make this true.
6. Write 2 expressions using properties of exponents that will be equivalent to x10
7. The LCM of two numbers is 24. Name as many possible number pairs that have a LCM of 24.
8. Find two numbers that equal 8 when you subtract them.
9. Find 2 three-digit numbers whose sum is 855.
10. Kendra added 2 integers. The sum was a negative number. What two numbers could she have
added?
11. Tobin subtracted two integers. His difference was -12. Find 3 pairs of numbers that he could have
subtracted.
12. Find 2 fractions whose quotient is 2 2/3.
13. Create an equation that has solution of x = 5. Use more than one operation.
14. Juan and five buddies attended the Captain America: Civil War premier at Keystone Cinemas. They
spent $97 total. They had purchased 5 bags of popcorn and 6 tickets. What could be the prices of the
tickets and popcorn in order to reach the $97 total?
Generalization
1. If the pattern shown in the table continues, what amount will have been raised by Week 5?
Week
Amount (thousands $)
Scholarship Funds
0
1
0
0
2
2
3
4
?
6
2. Which equation represents the table shown?
Hours
15
25
35
a. y = 8.5x
Money ($)
127.50
212.50
297.50
b. y = 8.5x + 12.50
c. y = 15x
d. y = 15x + 12.50
3. Describe the pattern you see in the list of numbers below:
13, 26, 39, 52, 65
4. Solve the expression 𝑥 2 ∙ 𝑥 5
5. Look at the original expression of 8 + 4 ∙ 2 + 6 ÷ 3. Below are three expressions that have the
same numbers but the parentheses are in different places. Determine which ones have the same
answer as the original expression and explain why they have the same answer.
a) 8 + (4 ∙ 2) + 6 ÷ 3
b) (8 + 4) ∙ 2 + 6 ÷ 3
c) 8 + 4 ∙ 2 + (6 ÷ 3)
6. Finish the pattern: 5, 10, 15, ___, 25, ___ . Write the rule.
7. What is the maximum number of digits you can get in the sum when you add 2 three-digit numbers?
Why?
8. What is the minimum number of digits you can get in the sum when you add 2 three-digit numbers?
Why?
9. Gemma said, “I found a way to decide if the difference is positive or a negative when I subtract
integers.” What way do you think Gemma found?
10. Find two integers that when subtracted have a positive number as the difference. How can you
come up with more integer pairs that fit the description?
11. Find two integers that when subtracted have a negative number as the difference. How can you
come up with more integer pairs that fit the description?
12. Find two integers that when subtracted have 0 as the difference. How can you come up with more
integer pairs that fit the description?
13. What do you notice about the quotient when you divide a fraction greater than 1 by a fraction that
is less than ½? Why?
14. What do you notice about the quotient when you divide a fraction less than ½ by a fraction that is
greater than 1? Why?
15. Look at the table below. Write a rule for the multiplication rule of integers from what you can see in
the table?
5 x -3
5 x -2
5 x -1
5x0
5x1
5x2
5x3
= -15
= -10
= -5
=0
=5
= 10
=15
-5 x -3
-5 x -2
-5 x -1
-5 x 0
-5 x 1
-5 x 2
-5 x 3
= 15
= 10
=5
=0
= -5
= -10
=-15
14. Juan and his buddies attended the Captain America: Civil War premier at Keystone Cinemas. They
spent a certain amount of money on popcorn, candy, and soda pop to share amongst them. The
following table shows their total after each friend purchased a ticket. How much did each ticket cost?
How much did they spend on snacks?
Number of Friends
4
5
6
7
Total Amount Spent
$73
$85
$97
$109
Flexibility
1. Describe how YOU would find the area of the figure given.
2. Solve the following showing all your work.
2(4x – 3 + 2x + 5) = 28
3. Create a geometry problem using multiplication of rational expressions to find an expression for the
area of a triangle. Then solve that problem.
4. Solve the following problem in 3 different ways by putting the parentheses in 3 different places.
Explain what relationship exists between all problems you created.
30 ÷ 3 𝑥 5
5. Use a property other than a product of powers to write an expression that is equivalent to 16x8.
6. Describe how to find the GCF of 12 and 36.
7. Use a whole number other than 1 and a fraction to get a product less than 1.
8. Add 458 and 397 in two different ways. How are the ways you added them alike? How are they
different?
9. Subtract -11 – (-8) = ?
-11 – (-9) = ?
-11 – (-20) = ?
10. Subtract -14 – (-15) = ?. Subtract it another way.
11. Divide 8/9 by 2/3 using two different methods. How are the methods you used to divide them alike?
How are they different?
12. How can you solve this equation in more than one way?
4(𝑥 + 3) − 6
−8=7
3
13. Juan and his buddies attended the Captain America: Civil War premier at Keystone cinemas. They
purchased 6 tickets and 5 bags of popcorn. The total they spent was $97. Juan’s sister, Christina,
detests Juan’s friends, so she and her friends went to see the movie in Louisville at Tinseltown. They
purchased 5 tickets and 4 bags of popcorn. They spent $80 total. The prices of the tickets and popcorn
were the same (somehow) at both theaters. How much did they cost?