4.1: Algebra Talk: Seeing Structure
Find a solution to each equation without writing anything down.
- $x + 1 = 5$
- $2(x+1) = 10$
- $3(x+1) = 15$
- $500 = 100(x+1)$
Let’s see how tape diagrams can help us answer questions about unknown amounts in stories.
Find a solution to each equation without writing anything down.
Draw a tape diagram to represent each situation. For some of the situations, you need to decide what to represent with a variable.
Each situation in the previous activity is represented by one of the equations.
While in New York City, is it a better deal for a group of friends to take a taxi or the subway to get from the Empire State Building to the Metropolitan Museum of Art? Explain your reasoning.
Many situations can be represented by equations. Writing an equation to represent a situation can help us express how quantities in the situation are related to each other, and can help us reason about unknown quantities whose value we want to know. Here are three situations:
An architect is drafting plans for a new supermarket. There will be a space 144 inches long for rows of nested shopping carts. The first cart is 34 inches long and each nested cart adds another 10 inches. The architect want to know how many shopping carts will fit in each row.
A bakery buys a large bag of sugar that has 34 cups. They use 10 cups to make some cookies. Then they use the rest of the bag to make 144 giant muffins. Their customers want to know how much sugar is in each muffin.
Kiran is trying to save \$144 to buy a new guitar. He has \$34 and is going to save \$10 a week from money he earns mowing lawns. He wants to know how many weeks it will take him to have enough money to buy the guitar.
We see the same three numbers in the situations: 10, 34, and 144. How could we represent each situation with an equation?
In the first situation, there is one shopping cart with length 34 and then an unknown number of carts with length 10. Similarly, Kiran has 34 dollars saved and then will save 10 each week for an unknown number of weeks. Both situations have one part of 34 and then equal parts of size 10 that all add together to 144. Their equation is $34+10x=144$.
Since it takes 11 groups of 10 to get from 34 to 144, the value of $x$ in these two situations is $(144-34)\div{10}$ or 11. There will be 11 shopping carts in each row, and it will take Kiran 11 weeks to raise the money for the guitar.
In the bakery situation, there is one part of 10 and then 144 equal parts of unknown size that all add together to 34. The equation is $10+144x=34$. Since 24 is needed to get from 10 to 34, the value of $x$ is $(34-10)\div{144}$ or $\frac16$. There is $\frac16$ cup of sugar in each giant muffin.