standard_description: Interpret multiplication as scaling (resizing), by:
- Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.
- Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.
problem_activity: Task
Amy and Greg look for a method to estimate the number of leaves on a $35$ foot tree in their yard. Greg notices that the tree blocks almost all of the sunlight beneath its leaves so he thinks of the following way to estimate how many leaves are on the tree:
We can first measure the area of the ground covered by the tree.
Then we measure the area of an average leaf.
We will need to estimate how much of its area an average leaf shades and how many leaves lie over an average point under the tree.
With all of this information we should be able to get a good estimate for the number of leaves on the tree.
Amy and Greg find that the tree covers a region which is roughly a circle $30$ feet in diameter. What is the approximate area covered by the tree?
How can Amy and Greg effectively measure an irregular shape such as a tree leaf?
How can Amy and Greg effectively decide what number to multiply by to account for multiple leaves lying over the same area and leaves shading less than their full surface area?
Suppose $a$ is the approximate area underneath the tree, $b$ the average leaf area, $c$ the proportion of its area an average leaf shades, and $d$ the average number of leaves found over a spot underneath the tree. In terms of the numbers $a$, $b$, $c$, and $d$ what formula will Amy and Greg find for the number of leaves on the tree?


aligns: no