Adding Fractions
We can express 5/6 as the sum of unit fractions in the form of 1 over a counting number either as 1/2 + 1/3 or as 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6. Express 7/12 as the sum of unit fractions without using 1/12.
1/3 + 1/4 is one solution, but there are others.
Determine 3,229 ÷ 25. Give your answer as a mixed-number and a decimal.
129 4/25
129.16
Jeff is 50 and Sarah is 10. In how many years will Jeff be twice as old as Sarah?
30 years
A regular figure has all the angles the same measure and all the side lengths the same as well. A regular triangle has angle measures 60 degrees. A regular quadrilateral (called a square!) has angle measures of 90 degrees. What are the angle measures of a regular pentagon? Hexagon? An n-sided regular figure?
Pentagon = 108 degrees
Hexagon = 120 degrees
n-sided figure = 180(n-2)/n
When you add 0.3 + 0.4 you get 0.7. Write two different numbers that have one decimal each that add to having 0 decimal places.
0.4 + 0.6
There are other solutions.
We can express 5/6 as the sum of unit fractions in the form of 1 over a counting number either as 1/2 + 1/3 or as 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6. Express 4/5 as the sum of unit fractions without using 1/5. (Hint: You will need at least 3 fractions.)
1/2 + 1/5 + 1/10 is one solution, but there are others.
There are two different fractions. In each fraction the difference between the numerator and the denominator is the same value (numerator minus the denominator in both cases). Assuming all the numbers are positive, how can you tell which fraction is larger?
The fraction with the larger numbers is larger.
Jenny is 3 times as old as David right now. In 8 years she will be twice as old as David. How old is each person now?
Jenny is 24, David is 8.
How many lines of symmetry do regular figures have? Triangle, quadrilateral, pentagon, hexagon, n-sided?
The number of lines of symmetry is the number of sides of the regular figure.
Write two numbers that each has two decimal places that the sum has 0 decimal places. (Remember that 0.50 really only has one decimal place since it is equal to 0.5 so each of these decimals can't end in a zero.)
0.45 + 0.55
There are other solutions.
When adding 2/3 + 4/5 the sum can be expressed 2x5+4x3/3x5 = 22/15, which is in lowest terms. Under what circumstances of adding fractions in this manner will the sum absolutely not be in lowest terms?
When the denominators share a common factor.
I have a certain number of pieces of candy. I ate half on the first day, two-thirds of the remaining pieces on the second day and three-fourths on the third day. On the fourth day I ate the remaining 2 pieces. How many pieces did I start with?
Going backward- 2 was ¼ of what remained before eating ¾ so there were 8 before eating ¾. 8 was 1/3 remaining after eating 2/3 so before eating 2/3 there were 24. 24 was ½ of what was there before eating ½ so originally there were 48 pieces of candy.
3 and 5 are both prime. 5 and 7 are both prime. 11 and 13 are both prime. Find the next 3 sets of primes that differ by 2.
17,19 and 29,31, and 41,43
How many lines of symmetry does a circle have?
Infinite
We know that 1/2 is equivalent to 2/4 and 3/6 and 4/8. What other decimals are equivalent to .5?
.50, .500, etc.
We know that when we add fractions we need a common denominator, so adding 2/3 + 5/7 does not equal 7/10. (This is a very common mistake in adding fractions.) Find two fractions so that when you take half the correct sum, it is larger than the wrong sum which you get by adding the numerators and adding the denominators.
1/3 + 1/5
The average of the real sum is larger.
There are many others.
Find two different mixed numerals in the form of "a b/c" so that a x c + b is the same for both fractions. Find the difference between the two fractions and try to make the difference as small as possible.
5 3/4 and 4 3/5
There are many more answers, but anything where the denominator and the whole number part switch works.
We have seen that square numbers add the next odd number: 1, 1+3=4, 4+5=9 . . . and that these numbers can form a square. There are also triangular numbers that form a triangle and these numbers add the next integer: 1, 1+2=3, 3+3=6, 6+4=10. So the first 4 triangular numbers are 1, 3, 6, 10. After 1 what is the next number that is both a square number and a triangular number. The third number is in the thousands and the fourth number is more than 41,000.
36, 1225, 41616
The perpendicular bisector of a side of a triangle is a line that goes through the midpoint (the point that is exactly in the middle of a segment) and makes a right angle with the segement.
Draw an acute triangle and draw the 3 perpendicular bisectors and see where they intersect.
Do this for an obtuse triangle.
Do it for a right triangle and compare where the points of intersection are. (the perpendicular bisectors should all intersect at a point)
Where are they located for each type of triangle?
For an acute triangle the point is inside the triangle, for an obtuse triangle the point is outside the triangle, and for a right triangle the point is on the hypotenuse (the longest side, which is opposite the right angle).
8,000 ÷ 20 = 400 because 8÷2=4 and 3 zeros divided by 1 zero will be 2 zeros (you subtract the zeros). What is 24,000,000 ÷ 600?
40,000
Find two fractions so that the correct sum is greater than the wrong sum.
3/4 + 4/5
The wrong sum is larger. There are many other answers.
Subtract 1 - 1/2 -1/4 - 1/8 - 1/16 - 1/32 - 1/64.
1/64
8 > 4. If we multiply both 8 and 4 by -1 we get -8 < -4. Explain what happens to the inequality sign when you multiply by a negative.
It flips the inequality sign.
If we take a circle and make 3 cuts through the center we will have 6 sections. If we make 3 cuts that do not have to go through the center, what is the most number of sections we can make (the sections do not have to be the same shape).
7
We know that 100 ÷ 25 = 4 so how can you determine 3,200 ÷ 25?
32x4=128