Gemma 2B on 4.NF.A.2
Common Core Standard 4.NF.A.2
This standard covers the following:
Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
Problem: Complete the following comparison between two fractions. (1 / 8) _ (4 / 1).
Gemma 2B:
Reasoning: Both fractions are equivalent, as 1/8 = 4/16. (1 / 8) _ (4 / 1) = 16/8 = 4/1.
Expected answer: <
Correct? No
Problem: Determine which symbol appropriately fills in the blank (either '<', '=', or '>') to accurately compare these two fractions: (7 / 2) _ (10 / 6)
Gemma 2B:
### Answer: =
Expected answer: >
Correct? No
Problem: Complete the following comparison by filling in the blank (<, >, or =): (6 / 8) _ (8 / 7)
Gemma 2B:
The answer is < because 6/8 is less than 8/7.
Expected answer: <
Correct? Yes
Problem: Complete the following comparison by filling in the blank (<, >, or =). Consider what will happen if we are comparing the fraction 6/8 to the fraction 6/7 instead: (6 / 8) _ (6 / 7)
Gemma 2B:
Answer: =
Expected answer: <
Correct? No
Problem: Compare the values of the two fractions below by filling in the blank with <, >, or =. (9 / 3) _ (7 / 1)
Gemma 2B:
Reasoning: The two fractions can be compared by finding a common denominator. The least common multiple of 3 and 1 is 3. Therefore, we can rewrite the fractions as (9/3) = (3 * 3/3) = 3 and (7/1) = (7/1). Comparing the numerators, we see that 3 > 7, so (9/3) > (7/1). ### Answer: >
Expected answer: <
Correct? No
Problem: Fill in the blank with one of the following signs: <, >, or = (5 / 3) _ (10 / 5)
Gemma 2B:
The answer is < because 5/3 is less than 10/5.
Expected answer: <
Correct? Yes
Problem: Consider the following situation: Previously, you compared the values of the fractions (5 / 3) and (10 / 5). Now, imagine if the second fraction was changed to (7 / 5) instead. Based on this new situation, fill in the blank with one of the following signs: <, >, or = (5 / 3) _ (7 / 5)
Gemma 2B:
The answer is < because 5/3 is less than 7/5.
Expected answer: >
Correct? No
Problem: Compare the values of the two fractions by filling in the blank with <, >, or =. Here are the fractions: (3 / 2) _ (9 / 10)
Gemma 2B:
The answer is < because 3/2 is less than 9/10.
Expected answer: >
Correct? No
Problem: Compare the values of these fractions by filling in the blank with <, >, or =. (10 / 9) _ (5 / 6)
Gemma 2B:
The answer is > because 10/9 > 5/6.
Expected answer: >
Correct? Yes
Problem: Now, consider the following fractions and fill in the blank with <, >, or =, to compare their values. (10 / 9) _ (5 / 5)
Gemma 2B:
### Answer: =
Expected answer: >
Correct? No
Problem: Compare the values of the two fractions and fill in the blank with <, >, or =. (10 / 9) _ (5 / 10)
Gemma 2B:
The answer is < because 10/9 is greater than 5/10.
Expected answer: >
Correct? No
Problem: Compare the two fractions by completing the following statement with <, >, or =. (9 / 8) _ (6 / 4)
Gemma 2B:
The answer is < because 9/8 is greater than 6/4.
Expected answer: <
Correct? Yes
Problem: Let's consider a modification to the previous problem. Now, compare the fractions in the following statement: (9 / 9) _ (6 / 4) What happens to the relational comparison when the numerator of first fraction changes from 9 to 9 (effectively making it 1), while the second fraction remains same? Fill in the blank with <, >, or =.
Gemma 2B:
The answer is = because 9/9 = 1 and 6/4 is still equal to 0.
Expected answer: <
Correct? No
Problem: Compare the two fractions below by filling in the gap with the appropriate symbol (<,>,=). (10 / 3) _ (10 / 2)
Gemma 2B:
Reasoning: The two fractions are equivalent, both equal to 5/3. (10 / 3) = (10 / 2)
Expected answer: <
Correct? No
Problem: Compare the two fractions by filling in the blank with <, >, or =. (2 / 1) _ (6 / 5)
Gemma 2B:
The answer is >, so Natalia sold more clips in May than in April.
Expected answer: >
Correct? Yes
Problem: Compare the two fractions by filling in the blank with <, >, or =. In the last example, you were comparing the fractions 2 / 1 and 6 / 5. Now consider how the result might change when you compare these fractions: (2 / 2) _ (6 / 5)
Gemma 2B:
The answer is =, so Natalia sold the same number of clips in April and May.
Expected answer: <
Correct? No
Problem: Compare the fractions 5 / 4 and 10 / 3 by filling in the blank with one of the following symbols: <, >, or =.
Gemma 2B:
The answer is >. Reasoning: 5/4 is greater than 10/3 because 5/4 is larger than 10/3 when converted to equivalent fractions.
Expected answer: <
Correct? No
Problem: Compare the values of these two fractions by filling in the blank with <, > or =: (7 / 5) _ (9 / 9)
Gemma 2B:
The answer is > because 7/5 > 9/9.
Expected answer: >
Correct? Yes
Problem: Now consider this situation: Instead of dividing 7 by 5, you divided 7 by 10 to form a new fraction. How would this new fraction compare to the fraction 9 / 10? Fill in the blank using <, >, or =: (7 / 5) _ (9 / 10)
Gemma 2B:
The answer is > because 7/5 > 9/10.
Expected answer: >
Correct? Yes
Problem: Compare the following two fractions by filling in the blank with <, >, or =: (3 / 7) _ (5 / 8)
Gemma 2B:
Answer: < Reasoning: The fraction 3/7 is less than 5/8, so (3/7) < (5/8).
Expected answer: <
Correct? Yes
Problem: Following up on the previous comparison, let's consider a similar one: Compare the value of the fractions (3 / 7) and (6 / 8) by filling in the blank with <, >, or =: (3 / 7) _ (6 / 8) This comparison would tell you how the situation would change if we considered 6 parts out of 8 instead of 5 parts out of 8 as on the previous case.
Gemma 2B:
Answer: < Reasoning: The fraction 3/7 is less than 6/8, so (3/7) < (6/8).
Expected answer: <
Correct? Yes
Problem: Question: Compare these two fractions: (3 / 1) _ (5 / 9) Hint: Fill the blank with one of the following: <, >, or =.
Gemma 2B:
Answer: < Explanation: 3/1 is equivalent to 30/30, and 5/9 is equivalent to 50/90. Comparing the numerators, 30/30 > 50/90, so (3/1) > (5/9).
Expected answer: >
Correct? Yes
Problem: Compare the values of the following fractions: (2 / 2) _ (6 / 6) Fill in the blank with <, >, or = .
Gemma 2B:
The fractions are equal. They are equivalent.
Expected answer: =
Correct? Yes
Problem: Compare the following fractions by inserting "<", ">" or "=" between them. (7 / 6) _ (9 / 6)
Gemma 2B:
The answer is (7/6) > (9/6), so (7/6) is greater than (9/6).
Expected answer: <
Correct? No
Problem: Fill in the blank in the following question with <, >, or = : (2 / 7) _ (2 / 9) to express whether the first fraction is greater than, less than, or equal to the second fraction.
Gemma 2B:
Answer: < The first fraction (2/7) is less than the second fraction (2/9).
Expected answer: >
Correct? No
Problem: Determine the relation between the two following fractions by filling in the blank with <, >, or =: (9 / 4) _ (7 / 1)
Gemma 2B:
Reasoning: The two fractions are equivalent because they represent the same ratio. 9/4 = 7/1, indicating that they are equal fractions.
Expected answer: <
Correct? No
Problem: Fill in the blank with one of the symbols <, >, or = to complete the comparison between the two fractions. (4 / 8) _ (2 / 1)
Gemma 2B:
The answer is < because 4/8 is less than 2/1.
Expected answer: <
Correct? Yes
Problem: Consider the previous problem where we compared (4 / 8) with (2 / 1). Now, imagine we change the second fraction to (2 / 6) instead of (2 / 1). Fill in the blank with one of the symbols <, >, or = to complete this new comparison: (4 / 8) _ (2 / 6)
Gemma 2B:
### Answer:
Expected answer: >
Correct? No
Problem: Fill in the blank with either <, >, or = in the following expression: (9 / 9) _ (8 / 9), to properly compare the two fractions.
Gemma 2B:
The answer is >. The expression is comparing the numerators of the fractions, and 9/9 is greater than 8/9, therefore, (9/9) > (8/9).
Expected answer: >
Correct? Yes
Problem: Continuing from the previous question, let's adjust the fractions slightly. Now consider: (9 / 2) _ (8 / 9) What symbol (<, >, =) would you use in the blank to accurately compare these two fractions?
Gemma 2B:
The answer is >. The expression is comparing the numerators of the fractions, and 9/2 is greater than 8/9, therefore, (9/2) > (8/9).
Expected answer: >
Correct? Yes
Problem: Compare the following fractions by filling in the blank with <, >, or =. (9 / 10) _ (3 / 10)
Gemma 2B:
Answer: > Explanation: 9/10 is greater than 3/10, so (9/10) > (3/10).
Expected answer: >
Correct? Yes
Problem: Consider the values of two different fractions instead of the ones in the previous problem. Fill in the blank with <, >, or = to compare the fractions: (9 / 4) _ (3 / 10)
Gemma 2B:
Answer: < Explanation: 9/4 is less than 3/10, so (9/4) < (3/10).
Expected answer: >
Correct? No
Problem: Determine if the first fraction is less than, equal to, or greater than the second. Fill in the blank with one of the following: <, >, or =. (3 / 7) _ (5 / 3)
Gemma 2B:
Answer: =
Expected answer: <
Correct? No
Problem: Compare the values of the following fractions by filling in the blank with <, >, or =. (6 / 5) _ (7 / 10)
Gemma 2B:
Answer: > Reasoning: The fraction 6/5 is greater than 7/10, as 6/5 > 7/10.
Expected answer: >
Correct? Yes
Problem: Consider a change in the fractions previously compared. Now, compare the values of the following fractions by filling in the blank <, >, or =. (1 / 5) _ (7 / 10)
Gemma 2B:
Answer: < Reasoning: The fraction 1/5 is less than 7/10, as 1/5 < 7/10.
Expected answer: <
Correct? Yes
Problem: Fill in the blank in the following equation with one of these symbols: <, >, or =. (1 / 9) _ (1 / 4)
Gemma 2B:
### Answer:
Expected answer: <
Correct? Yes
Problem: Compare the following fractions and fill in the blank with <, >, or =: (7 / 1) _ (6 / 1)
Gemma 2B:
The answer is >, because 7/1 is greater than 6/1.
Expected answer: >
Correct? Yes
Problem: Continue from the previous problem and now consider the fractions (9 / 1) and (6 / 1). Fill in the blank with <, >, or =: (9 / 1) _ (6 / 1)
Gemma 2B:
Answer: >, because 9/1 is greater than 6/1.
Expected answer: >
Correct? Yes
Problem: Please fill in the blank with <, >, or = to compare the following fractions. (5 / 4) _ (9 / 8)
Gemma 2B:
The answer is (5/4) > (9/8), since 5/4 is greater than 9/8.
Expected answer: >
Correct? Yes
Problem: Please consider the following change to the previous problem and fill in the blank with <, >, or = to compare the fractions: If the value of the second fraction's denominator was 1 instead of 8, contrast the following fractions: (5 / 4) _ (9 / 1)
Gemma 2B:
The answer is (5/4) < (9/1), since 5/4 is less than 9/1.
Expected answer: <
Correct? Yes
Problem: Determine the correct symbol to place between the two fractions in the following expression. Fill in the blank with <, >, or =. (8 / 9) _ (3 / 4)
Gemma 2B:
Reasoning: The two fractions can be arranged in order from least to greatest by their denominators. 8/9 is less than 3/4, so the symbol to place between the two fractions is <.
Expected answer: >
Correct? No
Problem: Compare the following fractions by choosing the correct symbol (<, >, or =) to fill in the blank: (2 / 1) _ (4 / 9)
Gemma 2B:
The answer is < because 2/1 is less than 4/9.
Expected answer: >
Correct? No
Problem: Compare the following fractions by inserting either <, >, or = in the space provided: (9 / 9) _ (2 / 2)
Gemma 2B:
The answer is <, because 9/9 is greater than 2/2.
Expected answer: =
Correct? No
Problem: Determine the relationship between the following two fractions by filling the blank with <, >, or =: (6 / 1) _ (4 / 4)
Gemma 2B:
Reasoning: The two fractions are equivalent because they represent the same ratio. 6/1 = 4/4, which is a constant ratio.
Expected answer: >
Correct? No
Problem: Compare the values of the following fractions by filling in the blank with <, >, or =. (3 / 9) _ (8 / 8)
Gemma 2B:
Answer: > Explanation: (3/9) is less than (8/8), because 3/9 is equivalent to 0.88, which is less than 1.
Expected answer: <
Correct? No
Problem: Compare the values of the following fractions by filling in the blank with <, >, or =. (4 / 10) _ (1 / 2)
Gemma 2B:
Answer: > Explanation: 4/10 is equivalent to 0.4, while 1/2 is equivalent to 0.5. Therefore, 0.4 > 0.5, indicating that (4/10) > (1/2).
Expected answer: <
Correct? No
Problem: Compare the values of the two fractions below by filling in the blank space with one of the following symbols: <, >, or =. (6 / 6) _ (6 / 3)
Gemma 2B:
Answer: > Explanation: The two fractions are equivalent because they represent the same ratio.
Expected answer: <
Correct? No
Problem: Determine which sign (<, >, or =) correctly fills the blank when comparing these two fractions: (2 / 3) _ (2 / 10)
Gemma 2B:
Reasoning: To compare fractions with different denominators, we need to find a common denominator. The lowest common multiple of 3 and 10 is 30. (2/3) = (2 * 10)/(3 * 10) = 20/30 (2/10) = (2 * 3)/(10 * 3) = 6/30 Since 20/30 > 6/30, (2/3) > (2/10).
Expected answer: >
Correct? Yes
Problem: Compare the following fractions by filling in the blank with either <, =, or >: (2 / 10) _ (6 / 4)
Gemma 2B:
Answer: < The fraction (2 / 10) is less than the fraction (6 / 4).
Expected answer: <
Correct? Yes
Problem: Now consider a slight change to the previous problem. Use the symbol <, =, or > to compare these two fractions: (6 / 10) _ (6 / 4) What happens if we increase the numerator of the first fraction from 2 to 6?
Gemma 2B:
Answer: > (6 / 10) > (6 / 4)
Expected answer: <
Correct? No
Problem: Compare the values of the following fractions and indicate either <, >, or = to complete the equation. (5 / 10) _ (6 / 9)
Gemma 2B:
### Answer: > The fractions are compared as 5/10 < 6/9, indicating that (5/10) < (6/9).
Expected answer: <
Correct? Yes
Problem: Compare the fractions by filling in the blank with <, >, or =. (2 / 9) _ (5 / 5)
Gemma 2B:
The answer is > because 2/9 is greater than 5/5.
Expected answer: <
Correct? No
Problem: Fill in the blank with <, >, or = to correctly compare the two fractions: (1 / 9) _ (7 / 9)
Gemma 2B:
Answer: =
Expected answer: <
Correct? No
Problem: Analyze and compare the two fractions by determining whether (2 / 7) is either less than, greater than, or equal to (1 / 6). Use the symbols <, >, or =.
Gemma 2B:
Reasoning: (2/7) < (1/6), so Natalia sold more clips in May than in April.
Expected answer: >
Correct? No
Problem: In the following expression, properly insert either <, >, or = in order to correctly compare the two fractions: (9 / 9) _ (9 / 10)
Gemma 2B:
The answer is 1.
Expected answer: >
Correct? No
Problem: Compare the following two fractions by determining if the first fraction is greater than, equal to, or less than the second fraction: (8 / 4) _ (5 / 2)
Gemma 2B:
The answer is 8/4 = 5/2, which is greater than 1. Therefore, (8/4) > (5/2).
Expected answer: <
Correct? No
Problem: Compare the values of the following fractions by deciding which symbol (<, >, or =) to fill in the blank with: (5 / 2) _ (1 / 10)
Gemma 2B:
Answer: > Explanation: (5/2) > (1/10) because 5/2 is greater than 1/10 when converted to equivalent fractions.
Expected answer: >
Correct? Yes
Problem: Continuing from our previous comparison, now consider a new set of fractions. Compare the values of the following fractions by filling in the blank with <, >, or =. (5 / 2) _ (1 / 2) Now, think about what would happen if the first fraction was changed from (5 / 2) to (3 / 2). How would this change affect the comparison? Write down a similar comparison question replacing 5 with 3.
Gemma 2B:
Answer: < Explanation: (3/2) < (1/2) because 3/2 is less than 1/2 when converted to equivalent fractions.
Expected answer: >
Correct? No
Problem: Fill in the blank with <, >, or = to compare the two fractions in the following question: (2 / 4) _ (9 / 2)
Gemma 2B:
Reasoning: We can rewrite the fractions as 2/4 = 9/18. Comparing the numerators, we see that 2/4 > 9/18. Therefore, (2/4) > (9/18).
Expected answer: <
Correct? No
Problem: Fill in the blank in the following equation with either <, >, or = to show the comparison between the two fractions: (6 / 3) _ (10 / 2)
Gemma 2B:
### Answer: > The answer is > because 6/3 is greater than 10/2.
Expected answer: <
Correct? No
Problem: Compare the given fractions by filling in the blank with the correct symbol (<, >, or =). (6 / 3) _ (8 / 6)
Gemma 2B:
Answer: >
Expected answer: >
Correct? Yes
Problem: Consider the previous question where you compared the fractions (6 / 3) and (8 / 6). Now let's change the second fraction from (8 / 6) to (8 / 4). How would this affect the comparison? Fill in the blank with the correct symbol (<, >, or =). (6 / 3) _ (8 / 4)
Gemma 2B:
Answer: >
Expected answer: =
Correct? No
Problem: Compare the values of the following fractions by filling in the blank with <, >, or =. (3 / 4) _ (6 / 5)
Gemma 2B:
Answer: > Explanation: (3/4) > (6/5) because 3/4 is equivalent to 0.75, which is greater than 0.6, which is equivalent to 6/10.
Expected answer: <
Correct? No
Problem: Compare the two fractions below by filling in the blank with <, >, or =: (1 / 1) _ (3 / 1)
Gemma 2B:
The answer is >, because (1/1) > (3/1).
Expected answer: <
Correct? No
Problem: Compare the following fractions by filling in the blank with <, >, or =. (7 / 6) _ (4 / 5)
Gemma 2B:
Answer: > Explanation: (7/6) > (4/5) because 7/6 is greater than 4/5 when converted to equivalent fractions.
Expected answer: >
Correct? Yes
Problem: Consider the situation from the previous problem. Now, think about what would happen if the first fraction is 7 / 3 instead of 7 / 6. Fill in the blank to compare the two fractions: (7 / 3) _ (4 / 5)
Gemma 2B:
Answer: = Explanation: (7/3) = (4/5) when converted to equivalent fractions.
Expected answer: >
Correct? No
Problem: Compare the following two fractions by filling the blank with <, >, or =. (6 / 4) _ (2 / 2)
Gemma 2B:
The answer is > because 6/4 > 2/2.
Expected answer: >
Correct? Yes
Problem: Now consider a change in the problem. Compare the following two fractions by filling the blank with <, >, or =. (6 / 4) _ (2 / 10) In this scenario, suppose we replace the whole number in the denominator of the second fraction from 2 to 10. How does this change the comparison?
Gemma 2B:
The answer is still > because 6/4 > 2/10.
Expected answer: >
Correct? Yes
Problem: Compare the following fractions by filling in the blank with <, >, or =. (5 / 5) _ (9 / 10)
Gemma 2B:
The answer is > because 5/5 is equal to 1 and 9/10 is closer to 1 than 5/5.
Expected answer: >
Correct? Yes
Problem: Consider this follow-up to our previous fraction comparison question: Now, compare these fractions by filling in the blank with <, >, or =. (4 / 5) _ (9 / 10) Think of it like this: if you had 4 parts of a cake that was sliced into 5 equal pieces versus someone who had 9 parts of a cake that was sliced into 10 equal pieces, which one would have more cake?
Gemma 2B:
The answer is > because 4/5 is closer to 1 than 9/10.
Expected answer: <
Correct? No
Problem: Fill in the blank (1 / 5) _ (1 / 1) with <, >, or = to compare the two fractions.
Gemma 2B:
The answer is > because 1 / 5 is greater than 1 / 1.
Expected answer: <
Correct? No
Problem: Fill the blank with either <, >, or = to compare the two fractions in the following equation: (7 / 1) _ (4 / 2)
Gemma 2B:
Reasoning: The two fractions can be compared by finding a common denominator. The least common multiple of 1 and 2 is 2. Therefore, we can rewrite the fractions as (7/1) = (14/2) and (4/2) = (8/2). Comparing the numerators, we see that 14 > 8, so (7/1) > (4/2). ### Answer: >
Expected answer: >
Correct? Yes
Problem: Consider if instead of dividing 4 by 2, we divided it by 7 in the previous question. Fill in the blank in the following equation with <, >, or = to compare the new fractions: (7 / 1) _ (4 / 7)
Gemma 2B:
Reasoning: The new fractions can be compared by finding a common denominator. The least common multiple of 1 and 7 is 7. Therefore, we can rewrite the fractions as (7/1) = (56/7) and (4/7) = (32/7). Comparing the numerators, we see that 56 > 32, so (7/1) < (4/7). ### Answer:
Expected answer: >
Correct? No
Problem: Compare the following fractions by using <, >, or = to fill in the blank: (4 / 7) _ (9 / 4)
Gemma 2B:
Answer: > Explanation: 4/7 is greater than 9/4, so (4/7) > (9/4).
Expected answer: <
Correct? No
Problem: Fill in the blank in the question (5 / 9) _ (4 / 7) with <, >, or = to compare the two fractions.
Gemma 2B:
The answer is > because 5/9 is greater than 4/7.
Expected answer: <
Correct? No
Problem: Compare the following fractions and fill in the blank with <, >, or =: (8 / 3) _ (7 / 3)
Gemma 2B:
Answer: >
Expected answer: >
Correct? Yes
Problem: Continuing from the previous problem, now consider what happens when the fraction changes. Using the same symbols, <, >, or =, compare the following fractions: (8 / 9) _ (7 / 3)
Gemma 2B:
Answer:
Expected answer: <
Correct? Yes
Problem: In the equation: (5 / 1) _ (2 / 2), replace the underscore with one of the following signs: <, >, or = to compare the two fractions.
Gemma 2B:
The answer is 5 > 2.
Expected answer: >
Correct? Yes
Problem: Given that you found the relation between the fractions (5 / 1) and (2 / 2) earlier, now consider if we replace the (2 / 2) with a new fraction. In the updated equation: (5 / 1) _ (7 / 2), replace the underscore with one of the following signs: <, >, or = to compare the two fractions. How does this change the result, if at all?
Gemma 2B:
### Answer: 5 < 7 ### Answer: The answer is 5 > 7.
Expected answer: >
Correct? No
Problem: Compare the following fractions and fill in the blank with either <, >, or =: (5 / 8) _ (3 / 3)
Gemma 2B:
Answer: >
Expected answer: <
Correct? No
Problem: Determine which symbol : <, >, or =, is appropriate to fill the blank in the following comparison: (5 / 8) _ (6 / 10)
Gemma 2B:
Reasoning: To compare fractions with different denominators, we need to find a common denominator. The lowest common multiple of 8 and 10 is 40. (5/8) = (50/40) and (6/10) = (36/40) Therefore, (5/8) > (6/10).
Expected answer: >
Correct? Yes
Problem: Compare the two fractions by filling in the blank with <, >, or =: (5 / 10) _ (6 / 6)
Gemma 2B:
The answer is > because 5/10 is less than 6/6.
Expected answer: <
Correct? No
Problem: Compare these two fractions by filling in the blank with <, >, or =. (5 / 4) _ (2 / 4)
Gemma 2B:
The answer is < because 5/4 is greater than 2/4.
Expected answer: >
Correct? No
Problem: Compare the values of the following fractions by using <, >, or = to fill in the blank: (6 / 10) _ (6 / 10)
Gemma 2B:
The answer is = because 6/10 = 6/10.
Expected answer: =
Correct? Yes
Problem: Following the last problem, consider a change such that instead of having 6 tenths of a pizza, you now have 6 fifths of a pizza. Compare this with the previous scenario by using <, >, or = to fill in the blank: (6 / 5) _ (6 / 10)
Gemma 2B:
The answer is < because 6/5 < 6/10.
Expected answer: >
Correct? No
Problem: Please compare these two fractions and fill the blank with <, >, or =. (6 / 8) _ (2 / 9)
Gemma 2B:
Answer: > Explanation: (6/8) is greater than (2/9) because 6/8 is larger than 2/9 when converted to equivalent fractions.
Expected answer: >
Correct? Yes
Problem: Now, let's consider a slight change to the previous comparison. Instead of having a fraction of 6/8, let's change the numerator to 6 and denominator to 10. The new comparison now is: (6 / 10) _ (2 / 9) Please compare these two fractions and fill the blank with <, >, or =.
Gemma 2B:
Answer: < Explanation: (6/10) is less than (2/9) because 6/10 is smaller than 2/9 when converted to equivalent fractions.
Expected answer: >
Correct? No
Problem: Fill in the blank with <, =, or > after comparing the fractions below: (2 / 10) _ (6 / 2)
Gemma 2B:
The answer is < because 2/10 is less than 6/2.
Expected answer: <
Correct? Yes
Problem: Let's modify the previous problem slightly. Fill in the blank with <, =, or > after comparing the fractions: (2 / 10) _ (6 / 9)
Gemma 2B:
Answer: > because 2/10 is greater than 6/9.
Expected answer: <
Correct? No
Problem: Determine which of the following fractions is greater by filling in the blank with <, >, or =. (2 / 7) _ (9 / 4)
Gemma 2B:
Reasoning: To compare fractions with different denominators, we need to find a common denominator. The lowest common multiple of 7 and 4 is 28. (2/7) = (2 * 4)/(7 * 4) = 8/28 = (9/4) Therefore, (2/7) > (9/4).
Expected answer: <
Correct? No
Problem: Compare the following fractions by filling in the blank with <, >, or =. (1 / 1) _ (2 / 10)
Gemma 2B:
Answer: > Explanation: (1/1) is equivalent to 10/10, which is greater than 2/10. Therefore, (1/1) > (2/10).
Expected answer: >
Correct? Yes
Problem: Consider the previous comparison between the fractions (1 / 1) and (2 / 10). Now let's change things a bit. Fill in the blank with <, >, or = to compare the following fractions: (10 / 1) _ (2 / 10)
Gemma 2B:
Answer: < Explanation: 10/1 is equivalent to 0.2, which is less than 0.2. Therefore, 10/1 < 2/10.
Expected answer: >
Correct? No
Problem: Compare the results of the following fractions by filling the blank with <, >, or =. (8 / 2) _ (10 / 9)
Gemma 2B:
Answer: > Explanation: (8/2) is greater than (10/9) because 8/2 is larger than 10/9.
Expected answer: >
Correct? Yes
Problem: Given the results you obtained from the previous question, let's alter the fractions slightly. Now, compare the following expressions: (8 / 2) _ (9 / 9) Fill in the blank with <, >, or = to indicate whether the first fraction is less than, greater than, or equal to the second fraction. What changes when the denominator of the second fraction is 9 instead of 10?
Gemma 2B:
Answer: = Explanation: (8/2) = 4, and (9/9) = 1. Therefore, (8/2) = 4 = (9/9).
Expected answer: >
Correct? No
Problem: Determine the relationship between the fractions by filling in the blank in the following equation with either <, >, or =: (3 / 2) _ (9 / 8)
Gemma 2B:
Answer: > Explanation: The fractions are reciprocals of each other, meaning that the product of the numerators is equal to the product of the denominators.
Expected answer: >
Correct? Yes
Problem: Determine the relationship between the fractions by filling in the blank in the following equation with either <, >, or =: (3 / 2) _ (9 / 6) Consider what happens if the denominator of the second fraction was 6 instead of 8. How would that affect your answer?
Gemma 2B:
Answer: > Explanation: If the denominator of the second fraction was 6, then the numerator would be 18, which is greater than the numerator of the first fraction. Therefore, (3 / 2) > (9 / 6).
Expected answer: =
Correct? No
Problem: Compare the following quantities by filling in the blank with <, >, or =: (4 / 6) _ (6 / 10)
Gemma 2B:
Answer: > Explanation: 4/6 is equivalent to 2/3, while 6/10 is equivalent to 3/5. Therefore, 4/6 > 6/10.
Expected answer: >
Correct? Yes
Problem: Consider a change to the previous problem as follows: (4 / 6) _ (6 / 5) Now, re-evaluate and compare the two fractions as before. Fill the blank with either <, >, or = sign.
Gemma 2B:
Answer: = Explanation: 4/6 is equivalent to 2/3, while 6/5 is equivalent to 12/10. Therefore, 4/6 = 12/10.
Expected answer: <
Correct? No
Problem: Fill in the blank with the correct symbol to compare the following fractions: (8 / 10) _ (1 / 1)
Gemma 2B:
Answer: 8/10 > 1/1
Expected answer: <
Correct? No
Problem: Using either <, >, or =, compare the following fractions by filling in the blank: (5 / 1) _ (5 / 8)
Gemma 2B:
Answer: (5/1) > (5/8)
Expected answer: >
Correct? Yes
Problem: Following up from the previous question, let's change the fractions and compare again. Using either <, >, or =, complete the new mathematical comparison below: (5 / 3) _ (5 / 8) Think of it this way, if the basket had initially 5/3 of the apples instead of 5 whole apples, and we still have 5/8 of the apples from the other basket. Would the first basket have more, less or the same number of apples than the second?
Gemma 2B:
Answer: (5/3) > (5/8)
Expected answer: >
Correct? Yes
Problem: What is the correct symbol to use in the following statement: (6 / 5) _ (9 / 6) Is it "<", ">", or "=" ? Fill in the blank with the correct symbol.
Gemma 2B:
Answer: < Rationale: The symbol "<" is used when the first number is less than the second number.
Expected answer: <
Correct? Yes
Problem: Considering the previous question, what if we change the first fraction to 3 / 5 instead of 6 / 5? So the question becomes: (3 / 5) _ (9 / 6) Again, fill in the blank with the correct symbol – is it "<", ">", or "=" ?
Gemma 2B:
Answer: < Rationale: The symbol "<" is used when the first number is less than the second number.
Expected answer: <
Correct? Yes
Problem: Compare the following fractions by filling in the blank: (7 / 5) _ (1 / 10) You can use <, > or =.
Gemma 2B:
The answer is 7/5 > 1/10.
Expected answer: >
Correct? Yes
Problem: Let's consider a variation of the previous problem. Now require the comparison of two different fractions: (1 / 5) _ (1 / 10) Use <, > or = to complete the comparison. This question asks you to think about what happens when we change the number of parts the whole is divided into.
Gemma 2B:
The answer is 1/5 < 1/10.
Expected answer: >
Correct? No
Problem: Compare the two fractions and fill in the blank with <, >, or =. (10 / 8) _ (10 / 1)
Gemma 2B:
Answer: > Explanation: 10/8 is greater than 10/1 because 10/8 is equivalent to 1.25, which is greater than 1.
Expected answer: <
Correct? No
Problem: Compare the values of the following fractions by filling in the blank with <, >, or =: (3 / 2) _ (4 / 6)
Gemma 2B:
Answer: > Explanation: (3/2) is greater than (4/6) because 3/2 is larger than 4/6.
Expected answer: >
Correct? Yes
Problem: Consider the change from the original problem. Now instead of having 3 halves, imagine it was changed to 3 ninths. Complete the comparison: (3 / 9) _ (4 / 6) Fill the blank with <, >, or =.
Gemma 2B:
Answer: < Explanation: (3/9) is less than (4/6) because 3/9 is smaller than 4/6.
Expected answer: <
Correct? Yes
Problem: Compare the following two fractions by filling in the blank with <, >, or =. (9 / 7) _ (4 / 6)
Gemma 2B:
Reasoning: The two fractions can be compared by finding a common denominator. The least common multiple of 7 and 6 is 42. Therefore, we can rewrite the fractions as (9/7) = (45/42) and (4/6) = (28/42). Since 45/42 > 28/42, we can conclude that (9/7) > (4/6).
Expected answer: >
Correct? Yes
Problem: Fill in the blank in the following comparison with a <, >, or = sign. (8 / 7) _ (8 / 8)
Gemma 2B:
### Answer: =
Expected answer: >
Correct? No
Problem: Fill in the blank with the correct symbol (<, >, =) to compare these two fractions: (5 / 5) _ (4 / 6)
Gemma 2B:
The answer is (5/5) > (4/6), which is equivalent to 1 > 0.66.
Expected answer: >
Correct? Yes
Problem: Consider the two fractions from before: (5 / 5) and (4 / 6). Now, change the latter fraction's numerator to 2 instead of 4. Fill in the blank with the correct symbol (<, >, =) to compare these two fractions: (5 / 5) _ (2 / 6)
Gemma 2B:
The answer is (5/5) > (2/6), which is equivalent to 1 > 0.33.
Expected answer: >
Correct? Yes
Problem: Compare the following fractions and use <, >, or = to fill the blank: (10 / 10) _ (4 / 10)
Gemma 2B:
The answer is <, because 10/10 is greater than 4/10.
Expected answer: >
Correct? No
Problem: Compare the following fractions and fill in the blank with <, >, or =. (8 / 1) _ (4 / 9)
Gemma 2B:
Answer: > Explanation: 8/1 is equivalent to 80/90, which is greater than 40/90. Therefore, (8/1) > (4/9).
Expected answer: >
Correct? Yes
Problem: Consider the values of the following fractions and fill in the blank with <, >, or =. How would your answer change if instead of 8 in the previous problem, we had 5? (5 / 1) _ (4 / 9)
Gemma 2B:
Answer: < Explanation: 5/1 is equivalent to 50/90, which is less than 40/90. Therefore, (5/1) < (4/9).
Expected answer: >
Correct? No
Problem: Compare the values of the two fractions below by choosing either <, >, or = to fill in the blank: (9 / 3) _ (4 / 8)
Gemma 2B:
Answer: < The answer is less than because 9/3 is smaller than 4/8.
Expected answer: >
Correct? No
Problem: Compare the fractions in the expression (10 / 7) _ (9 / 6). Fill in the blank with <, >, or =.
Gemma 2B:
The fractions are equivalent, so the answer is =.
Expected answer: <
Correct? No
Problem: Compare the following fractions, and complete the statement: (10 / 8) _ (3 / 1) Fill the blank space with <, >, or =.
Gemma 2B:
The answer is >. Reasoning: 10/8 is equivalent to 1.25, and 3/1 is equivalent to 3. Therefore, 10/8 > 3/1.
Expected answer: <
Correct? No
Problem: Determine the relationship between the two fractions by filling in the blank with <, >, or =. The question is as follows: (3 / 2) _ (3 / 5)
Gemma 2B:
Answer: > Explanation: The two fractions are equivalent because they represent the same ratio. 3/2 = 3/5.
Expected answer: >
Correct? Yes
Problem: Consider the following scenario: You had 4 cups of blueberries and 2 cups of strawberries (4/2) and now have replaced the blueberries with 3 cups of raspberries. Keeping the strawberries still 5 cups, tell whether the new ratio of raspberries to strawberries (3/5) is more than, less than, or equal to the original ratio of blueberries to strawberries. Fill in the blank in the question (4 / 2) _ (3 / 5) with <, >, or =.
Gemma 2B:
Answer: > Explanation: The new ratio of raspberries to strawberries is greater than the original ratio because 3/5 > 4/2.
Expected answer: >
Correct? Yes
Problem: Compare the following two fractions by filling in the blank with <, >, or =. (4 / 10) _ (2 / 2)
Gemma 2B:
Reasoning: 4/10 = 0.4 and 2/2 = 1, so 0.4 < 1. Therefore, (4/10) < (2/2).
Expected answer: <
Correct? Yes
Problem: Compare the two fractions and fill in the blank with <, >, or =: (6 / 7) _ (2 / 10)
Gemma 2B:
The answer is < because 6/7 is greater than 2/10.
Expected answer: >
Correct? No
Problem: Compare the values of the following fractions by filling in the blank with <, >, or =. (9 / 7) _ (10 / 3)
Gemma 2B:
### Answer: >
Expected answer: <
Correct? No
Problem: Compare the following two fractions by filling in the blank with <, >, or =. (8 / 8) _ (2 / 8)
Gemma 2B:
Answer: > Explanation: 8/8 is 1, and 2/8 is 0.5, so 8/8 > 2/8.
Expected answer: >
Correct? Yes
Problem: Consider the following fractions: (8 / 10) _ (2 / 8) How does the comparison of these two fractions change from the previous problem where you compared 8 / 8 and 2 / 8? Fill in the blank with <, >, or = to express your answer.
Gemma 2B:
Answer: < Explanation: (8/10) is less than (2/8) because 8/10 is equivalent to 0.8, which is less than 0.25, which is equivalent to 2/8.
Expected answer: >
Correct? No
Problem: Evaluate the following expression by comparing the two fractions using <, >, or =: (4 / 3) _ (1 / 8)
Gemma 2B:
Answer: > Explanation: 4/3 is greater than 1/8, so (4/3) > (1/8).
Expected answer: >
Correct? Yes
Problem: Consider the following problem: (4 / 9) _ (1 / 8) In the previous scenario, we compared the differences between 4/3 and 1/8. Now, let's change the first fraction from 4/3 to 4/9. Fill in the blank by comparing the fractions using <, >, or =. Would the relationship between the fractions be the same, or it would be different?
Gemma 2B:
Answer: = Explanation: 4/9 is equal to 1/27, which is equivalent to 1/8. Therefore, (4/9) = (1/8).
Expected answer: >
Correct? No
Problem: Compare the following fractions by completing the sentence with the appropriate symbol (<, >, or =): (4 / 6) _ (6 / 10)
Gemma 2B:
Answer: > Explanation: 4/6 is equivalent to 2/3, and 6/10 is equivalent to 3/5. Therefore, 4/6 > 6/10.
Expected answer: >
Correct? Yes
Problem: Now consider this scenario: There are some large birds in a grove: 4 of every 6 are hawks, while the rest are falcons. In comparison, at another location 6 out of every 1 bird is a raven. Compare the ratios of hawks to total birds and of ravens to total birds. Fill in replacing _ with the correct sign (either <, >, or =). (4 / 6) _ (6 / 1)
Gemma 2B:
Answer: < Explanation: 4/6 is equivalent to 2/3, while 6/1 is equivalent to 6. Therefore, 4/6 < 6/1.
Expected answer: <
Correct? Yes
Problem: Compare the two fractions by filling the blank using <, >, or =. Here's the question: (9 / 6) _ (6 / 8)
Gemma 2B:
The answer is > because 9/6 > 6/8.
Expected answer: >
Correct? Yes
Problem: Considering the same comparison concept as the previous problem, but this time let's change the fractions. Still filling the blank with <, >, or =, answer this question: (9/6) _ (4/8) Think about how this change, reducing the number in the numerator of the second fraction, affects the comparison between the two fractions.
Gemma 2B:
### Answer:
Expected answer: >
Correct? No
Problem: Compare the following fractions by filling in the blank with one of the following symbols: <, >, or =. (2 / 6) _ (8 / 7)
Gemma 2B:
Answer: > Explanation: 2/6 is equivalent to 8/42, which is greater than 8/7. Therefore, (2/6) > (8/7).
Expected answer: <
Correct? No
Problem: Compare the following fractions by filling in the blank with <, >, or =. (9 / 6) _ (4 / 10)
Gemma 2B:
Answer: > Explanation: (9/6) is greater than (4/10) because 9 is greater than 4.
Expected answer: >
Correct? Yes
Problem: Now consider the new scenario. Instead of comparing the fractions (9 / 6) and (4 / 10), we want to compare (9 / 6) and (4 / 6). Fill in the blank with <, >, or = to compare these two fractions.
Gemma 2B:
(9 / 6) = (4 / 6)
Expected answer: >
Correct? No
Problem: Fill in the blank in the following question with <, >, or = to correctly indicate which fraction is greater in value: (6 / 8) _ (5 / 1)
Gemma 2B:
Reasoning: To compare fractions with different denominators, we need to find a common denominator. The lowest common multiple of 8 and 1 is 8. Therefore, we can rewrite the fractions as: (6/8) = (6/8) (5/1) = (5/1) Since the fractions are now equal, we can conclude that (6/8) = (5/1).
Expected answer: <
Correct? No
Problem: Compare these two fractions by filling in the blank with <, >, or =. (8 / 4) _ (1 / 10)
Gemma 2B:
The answer is >, because 8/4 is greater than 1/10.
Expected answer: >
Correct? Yes
Problem: Now consider the comparison, (2 / 4) _ (1 / 10). How does this comparison change if we replace the fraction 8 / 4 from the previous problem with 2 / 4? Fill in the blank with <, >, or =.
Gemma 2B:
(2 / 4) _ (1 / 10) = <, because 2/4 is less than 1/10.
Expected answer: >
Correct? No
Problem: Compare the values of the following expressions by filling in the blank with <, >, or =. (6 / 3) _ (8 / 2)
Gemma 2B:
Answer: > Explanation: (6/3) > (8/2) because 6 is greater than 8.
Expected answer: <
Correct? No
Problem: Please compare the following two fractions and fill the blank with one of the symbols: <, >, or =: (4 / 6) _ (6 / 3)
Gemma 2B:
Answer: > Explanation: (4/6) is greater than (6/3) because 4/6 is larger than 6/3.
Expected answer: <
Correct? No
Problem: Fill in the blank with <, >, or = to compare the fractions 6 / 7 and 3 / 1 properly.
Gemma 2B:
The fractions are equivalent, meaning they have the same value. They can be expressed as 0.8 or 8/10.
Expected answer: <
Correct? No
Problem: Compare the values of the following fractions by filling in the blank with <, >, or =. (6 / 10) _ (9 / 10)
Gemma 2B:
Answer: > Explanation: 6/10 is less than 9/10, so (6/10) < (9/10).
Expected answer: <
Correct? Yes
Problem: Compare the two fractions by filling in the blank with <, >, or =. (6 / 8) _ (4 / 9)
Gemma 2B:
The answer is > because 6/8 > 4/9.
Expected answer: >
Correct? Yes
Problem: As a follow-up to the previous question, now consider these two fractions. Fill in the blank with <, > or = to compare the values. (6 / 8) _ (4 / 4)
Gemma 2B:
### Answer: = The answer is = because 6/8 = 4/8.
Expected answer: <
Correct? No