Embedded Files
Learning Targets
- I understand the terms proportional relationship and constant of proportionality.
- I can use a table to reason about two quantities that are in a proportional relationship.
Notes
Notes
If the ratios between two corresponding quantities are always equivalent, the relationship between the quantities is called a proportional relationship.
This table shows different amounts of milk and chocolate syrup. The ingredients in each row, when mixed together, would make a different total amount of chocolate milk, but these mixtures would all taste the same.
Notice that each row in the table shows a ratio of tablespoons of chocolate syrup to cups of milk that is equivalent to 4:1.
About the relationship between these quantities, we could say:
- The relationship between amount of chocolate syrup and amount of milk is proportional.
- The relationship between the amount of chocolate syrup and the amount of milk is a proportional relationship.
- The table represents a proportional relationship between the amount of chocolate syrup and amount of milk.
- The amount of milk is proportional to the amount of chocolate syrup.
We could multiply any value in the chocolate syrup column by ¼ to get the value in the milk column. We might call ¼ a unit rate, because ¼ cups of milk are needed for 1 tablespoon of chocolate syrup. We also say that ¼ is the constant of proportionality for this relationship. It tells us how many cups of milk we would need to mix with 1 tablespoon of chocolate syrup.
Vocabulary
Vocabulary
proportional relationship: If there is a positive constant k so that the quantities x and y are related by the equation y=kx, then we say that y and x are in a proportional relationship, and that y is proportional to x. The constant k is called the constant of proportionality.
Activities
Activities
2.1 Notice and Wonder: Paper Towels by the Case
2.1 Notice and Wonder: Paper Towels by the Case
Here is a table that shows how many rolls of paper towels a store receives when they order different numbers of cases.
What do you notice about the table? What do you wonder?
Some things to notice:
- To go from one row to another, multiply both columns by the same number.
- To find the number of rolls, multiply the number of cases by 12.
- There are 12 rolls in a case.
Some things to wonder:
- How much does a case cost?
- How many paper towels are on a roll?
- Why would you need 120 rolls of paper towels?
2.2 Feeding a Crowd
2.2 Feeding a Crowd
- A recipe says that 2 cups of dry rice will serve 6 people. Complete the table as you answer the questions. Be prepared to explain your reasoning.
- How many people will 10 cups of rice serve?
- How many cups of rice are needed to serve 45 people?
2. A recipe says that 6 spring rolls will serve 3 people. Complete the table.
Add To Your Notes
Add To Your Notes
What is a proportional relationship?
When two quantities are always in equivalent ratios. So the relationship between the number of cups of rice and the number of people is a proportional relationship. The number of cups of rice is proportional to the number of people.
2.3 Making Bread Dough
2.3 Making Bread Dough
A bakery uses 8 tablespoons of honey for every 10 cups of flour to make bread dough. Some days they bake bigger batches and some days they bake smaller batches, but they always use the same ratio of honey to flour. Complete the table as you answer the questions. Be prepared to explain your reasoning.
- How many cups of flour do they use with 20 tablespoons of honey?
- How many cups of flour do they use with 13 tablespoons of honey?
- How many tablespoons of honey do they use with 20 cups of flour?
- What is the proportional relationship represented by this table?
Note that even though you multiply by a different scale factor to go from row to row in the table, the unit rate is always the same. Rename this “the constant of proportionality.” Note that It can always be found by finding how much of the second quantity per one of the first quantity.
Add To Your Notes
Add To Your Notes
What is a constant of proportionality?
A constant ratio (unit rate) in any proportional relationship. It can always be found by finding how much of the second quantity per one of the first quantity. In the Bread Dough activity, the constant of proportionality was 1.25. We could multiply any amount of honey by 1.25 to find the amount of flour needed.
Summary
Summary
Page updated
Report abuse