Lower Elementary:
Question: Emily is walking down the sidewalk and notices that the tiles on the sidewalk are repeating in a pattern: red, orange, yellow, green, blue, and purple. What color will the 35 tile be?
Answer: Blue
Solution: There are 6 colors in the tile pattern. That means every 6 tiles the pattern repeats. We can count by 6s until we get close to 35 so make counting the pattern easier. 6, 12, 18, 24, 30. So the 30th color is purple. That means the 31st color is red, the 32nd color is orange, the 33rd color is yellow, the 34th color is green, and the 35th color is blue.
Upper Elementary:
Question: Henry drank 6 cups of water on Monday, 2 pints of water on Tuesday, and 1 quart of water on Wednesday. How much water, in pints, did Henry drink altogether?
Answer: 7 pints
Solution: In order to combine the amounts of water, the units have to have the same name. There are 2 cups in 1 pint, so there are 4 cups in 2 pints, and 6 cups in 3 pints. In 1 quart, there are 2 pints. So, we add 3 pints + 2 pints + 2 pints to get the total amount of water Henry drank. 3 + 2 +2 = 7 pints of water.
Middle School:
Question: Brandon can ride his bike at a speed of 10 miles per hour. What is his speed in feet per minute?
Answer: 880 feet per minute
Solution: In order to convert the units, we have to know how many feel are in a mile and how many seconds are in an hour. In one mile, there are 5,280 feet. So, in 10 miles, there are 52,800 feet. There are 60 seconds in a minute. If we substitute 52,800 feet for 10 miles and 60 seconds for 1 hour, we have 52,800 feet / 60 minutes. Now let’s reduce the fraction. One way to reduce it is to reduce the top and bottom by 10 first to get 5,280 feet/6 minutes. Next we divide the top and bottom by 6. This gives us 880 feet per minute.
Algebra and Up:
Question: What is the equation of the line that goes through the point (–3, 2) and is perpendicular to the line 3x + 4y = 12?
Answer: y = 4/3x + 6
Solution: One way to solve this problem is to first find the slope of the line and then use slope-intercept form to find the equation of the line. The slope is perpendicular to the line 3x + 4y = 12. So first we need to find the slope of the line 3x + 4y = 12. One way to do this is to rewrite the equation into slope-intercept form.
3x + 4y = 12
Subtract 3x from both sides.
4y = –3x + 12
Divide both sides by 4.
y = –3/4x + 3
So the slope of this line is –3/4. Since the slope of the line we are trying to find is perpendicular to this line, we take the negative reciprocal of the slope. The negative reciprocal of –3/4 is 4/3. So, we have the slope of the line, 4/3, and a point that it goes through, (–3, 2). Let’s use slope-intercept form to find the equation of the line.
y = mx + b
Substitute the slope for m and the point for x and y so that we can solve for b.
2 = 4/3(–3) + b
Multiply 4/3 and –3
2 = –4 + b
Add –4 to both sides.
6 = b
The y-intercept of the line is 6. So the equation of the line is y = 4/3x + 6.
