4.1.1: Number Puzzles

Lesson

Let's solve some puzzles!

Exercise \(\PageIndex{1}\): Notice and Wonder: A Number Line

What do you notice? What do you wonder?

Figure \(\PageIndex{1}\): A number line from negative 5 to 14, by 1s. A blue arrow, pointing to the left, goes from 0 to negative 4. A red arrow, pointing to the right, goes from negative 4 to 3. A black arrow, pointing to the right, goes from 0 to 12. A green arrow, pointing to the right, goes from 3 to 12. A point is plotted on the number line at 12.

Exercise \(\PageIndex{2}\): Telling Temperatures

Solve each puzzle. Show your thinking. Organize it so it can be followed by others.

1. The temperature was very cold. Then the temperature doubled.
   Then the temperature dropped by 10 degrees. Then the temperature increased by 40 degrees. The temperature is now 16 degrees. What was the starting temperature?
2. Lin ran twice as far as Diego. Diego ran 300 m farther than Jada. Jada ran \(\frac{1}{3}\) the distance that Noah ran. Noah ran 1200 m. How far did Lin run?

Exercise \(\PageIndex{3}\): Making a Puzzle

Write another number puzzle with at least three steps. On a different piece of paper, write a solution to your puzzle.

Trade puzzles with your partner and solve theirs. Make sure to show your thinking.

With your partner, compare your solutions to each puzzle. Did they solve them the same way you did? Be prepared to share with the class which solution strategy you like best.

Are you ready for more?

Here is a number puzzle that uses math. Some might call it a magic trick!

1. Think of a number.
2. Double the number.
3. Add 9.
4. Subtract 3.
5. Divide by 2.
6. Subtract the number you started with.
7. The answer should be 3.

Why does this always work? Can you think of a different number
puzzle that uses math (like this one) that will always result in 5?

Summary

Here is an example of a puzzle problem: Twice a number plus 4 is 18. What is the number?

There are many different ways to represent and solve puzzle problems.

• We can reason through it.
  
  Twice a number plus 4 is 18.
  Then twice the number is \(18-4=14\).
  That means the number is 7.
  
• We can draw a diagram.

Figure \(\PageIndex{2}\): Three tape diagrams. First diagram is 3 parts labeled x, x, 4. Below it, a tape diagram of equal length with 1 part labeled 18. Second diagram is 2 parts labeled x, x. Below it, a tape diagram of the same width with 1 part labeled 14. Third diagram is 1 part labeled x. Below it, a tape diagram of equal length, 1 part, labeled 7.

• We can write and solve an equation, \(2x+4=18\) \(2x=14x=7\)

Reasoning and diagrams help us see what is going on and why the answer is what it is. But as number puzzles and story problems get more complex, those methods get harder, and equations get more and more helpful. We will use different kinds of diagrams to help us understand problems and strategies in future lessons, but we will also see the power of writing and solving equations to answer increasingly more complex mathematical problems.

Practice

Exercise \(\PageIndex{4}\)

Tyler reads \(\frac{2}{15}\) of a book on Monday, \(\frac{1}{3}\) of it on Tuesday, \(\frac{2}{9}\) of it on Wednesday, and \(\frac{3}{4}\) of the remainder on Thursday. If he still has 14 pages left to read on Friday, how many pages are there in the book?

Exercise \(\PageIndex{5}\)

Clare asks Andre to play the following number puzzle:

• Pick a number
• Add 2
• Multiply by 3
• Subtract 7
• Add your original number

Andre’s final result is 27.
Which number did he start with?

Exercise \(\PageIndex{6}\)

In a basketball game, Elena scores twice as many points as Tyler. Tyler scores four points fewer than Noah, and Noah scores three times as many points as Mai. If Mai scores 5 points, how many points did Elena score? Explain your reasoning.

Exercise \(\PageIndex{7}\)

Select all of the given points in the coordinate plane that lie on the graph of the linear equation \(4x-y=3\).

1. \((-1,-7)\)
2. \((0,3)\)
3. \((\frac{3}{4},0)\)
4. \((1,1)\)
5. \((2,5)\)
6. \((4,-1)\)

(From Unit 3.4.1)

Exercise \(\PageIndex{8}\)

A store is designing the space for rows of nested shopping carts. Each row has a starting cart that is 4 feet long, followed by the nested carts (so 0 nested carts means there's just the starting cart). The store measured a row of 13 nested carts to be 23.5 feet long, and a row of 18 nested carts to be 31 feet long.

Figure \(\PageIndex{3}\)

1. Create a graph of the situation.
2. How much does each nested cart add to the length of the row? Explain your reasoning.
3. If the store design allows for 43 feet for each row, how many total carts fit in a row?

(From Unit 3.2.1)

Exercise \(\PageIndex{9}\)

Triangle \(A\) is an isosceles triangle with two angles of measure \(x\) degrees and one angle of measure \(y\) degrees.

1. Find three combinations of \(x\) and \(y\) that make this sentence true.
2. Write an equation relating \(x\) and \(y\).
3. If you were to sketch the graph of this linear equation, what would its slope be? How can you interpret the slope in the context of the triangle?

(From Unit 3.4.2)

Exercise \(\PageIndex{10}\)

Consider the following graphs of linear equations. Decide which line has a positive slope, and which has a negative slope. Then calculate each line’s exact slope.

Figure \(\PageIndex{4}\): Graph of two lines, l and m, origin O, with grid. Horizontal axis, scale 0 to 12, by 1’s. Vertical axis, scale -80 to 140, by 20’s. Line l begins at 0 comma -80, passes through 8 comma 80. Line m begins at 0 comma 120, passes through 8 comma -40.

(From Unit 3.3.2)