Answer:
1. 8 slices left.
2. 1/10, 1/9, 1/4, and 1/3.
3. 2/3, 3/4, 8/9, 10/11
4. it's ratio should be propertional
Explanation:
1.
If there is two-thirds of a pizza left and the original pizza had a total of 3 slices, then there are 2 slices left because 2/3 of 3 is equal to 2.
If the original pizza had 12 slices, then two-thirds of it would be 8 slices (12 x 2/3 = 8), so there would be 4 slices left.
If the original pizza had 24 slices, then two-thirds of it would be 16 slices (24 x 2/3 = 16), so there would be 8 slices left.
2.
To order fractions from least to greatest, we need to find a common denominator.
The denominators are already small, so we can find the least common multiple (LCM) by inspection.
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, ...
Multiples of 4: 4, 8, 12, 16, 20, 24, ...
Multiples of 9: 9, 18, 27, 36, ...
Multiples of 10: 10, 20, 30, 40, ...
We can see that the least common multiple of 3, 4, 9, and 10 is 180.
1/3 = 60/180
1/4 = 45/180
1/9 = 20/180
1/10 = 18/180
Now that we have converted the fractions to have a common denominator, we can compare them.
1/10 is the smallest fraction since its numerator is the smallest.
Next, we have 1/9 and 1/4. Since the denominators are the same, we can simply compare the numerators.
1/9 is smaller than 1/4, so we can write them in order as:
1/10, 1/9, 1/4
Finally, we have 1/3.
1/3 is bigger than all the other fractions, so the final order is:
1/10, 1/9, 1/4, 1/3
Therefore, the fractions from least to greatest are 1/10, 1/9, 1/4, and 1/3.
3.
To place these fractions in order from least to greatest, we can compare them by finding a common denominator.
Step 1: Find a common denominator for all the fractions. The least common multiple (LCM) of 3, 4, 9, and 11 is 396, so we can convert each fraction into an equivalent fraction with a denominator of 396.
2/3 = (2/3) x (132/132) = 264/396
3/4 = (3/4) x (99/99) = 297/396
8/9 = (8/9) x (44/44) = 352/396
10/11 = (10/11) x (36/36) = 360/396
Step 2: Arrange the fractions in order from least to greatest based on their values.
To do this, we can compare the fractions by finding their equivalent decimal values using division.
2/3 = 0.666666...
3/4 = 0.75
8/9 = 0.888888...
10/11 = 0.909090...
From this comparison, we can see that the fractions are in order from least to greatest: 2/3, 3/4, 8/9, 10/11
4.
The student's argument that 2/4 = 2/3 because both fractions have the same number of dots filled is incorrect. Even though both fractions have a numerator of 2, they have different denominators, which means they represent different amounts.
To help the student understand this concept, you could use a visual representation, such as drawing two rectangles of different sizes, with four and three dots respectively, and fill in two dots in each rectangle. Then you could point out that even though they have the same number of dots filled in, the rectangles are different sizes and therefore the fractions are different. You could also explain that to compare fractions, we need to find a common denominator, and that the fractions 2/4 and 2/3 are not equivalent because they have different denominators. Finally, you could offer additional practice problems to help the student reinforce their understanding of fractions and equivalent fractions.