Tutor profile: Sean A.
To develop a deeper understanding of area and perimeter, I like the following questions: 1. Find the dimensions of two rectangles (or another shape) where the first has a larger perimeter and the second a larger area. 2. Find the dimensions of two rectangles (or another shape) where one the first is exactly twice the perimeter and the second is exactly twice the area.
The first question can be solved by using inequalities or by trial-and-error. This will lead a student to recognize that as a rectangle is stretched out (the difference between the length and width increases), the perimeter fits less area (so to speak). There are unlimited answers, but an example would be a 7x8 (perimeter=32 and area=56) and 4x13 rectangles (perimeter=34 and area=52). The second equation is a great extension as it is more challenging to solve using trial-and-error. It can be solved using a system of equations which will lead to a quadratic equation as well.
How tall is the tallest tree (or another object that is too tall to safely measure) outside your home? An extension after this first question has been solved is to have the student build a LEGO tree that is proportional to the original tree. The student will follow the same scale factor between their height and a LEGO mini-figure.
Proportions are a central concept in pre-algebra and the topic lends itself to some fun and interesting problems and activities. Finding the height of an object outside that is too tall to safely measure (e.g. a tall tree) is to use similar triangles formed by the object and its shadow. On a sunny day, the student will measure the length of the tall object's shadow. Then to create a similar triangle, the student will measure their height and their shadow right after measuring the object (to ensure the same angle from the sun). Then the height of the tree can be found using proportions. For example, if a tree's shadow is 432 inches, the person's height is 70 inches and their shadow is 78 inches, then the proportion will be: tree's height/432 = 70/78 Solving for the tree's height (using cross-multiply, scale factor, or isolating the variable) will result in a tree that is about 388 feet or about 32 1/3 feet.
A favorite problem for introducing and/or digging into systems of linear equations: 1. Choose three numbers 2. Find all the totals when added in pairs. (e.g. if the numbers are 13, 7 and 18, the totals will be 20, 25 and 31) 3. Exchange these three totals with someone 4. Figure out the original three values
This is a great opportunity to gain insight into a student's problem-solving strategies. After a student has their solution or best guess, I invite them to share their strategies and thought process. Eventually, we will complete the activity with a system of equations and solving using either substitution or elimination. For example: x + y = 20 y + z = 25 x + z = 31 Using substitution: Solve the first equation for x: x = 20 - y Solve the second equation for z: z = 25 - y Substitute for x and z in the third equation: 20 - y + 25 - y = 31 Solve for y: y = 7 Substitute y = 7 into the first two equations to find x = 13 and z = 18
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