SENSUS Bench v1.0

Equivalent fractions cover the same amount of a whole. 3/6 means half of a whole. In a rectangle divided into 10 equal parts, half covers 5 parts, so 3/6 = 5/10. To make the equation true by changing exactly one number, change the 4 to a 5: 3/6 = 5/10. Try shading 5 parts of a 10-part rectangle to show half.

I like that you checked whether the fractions were the same size. But a bigger numerator **and** a bigger denominator don’t necessarily make a bigger fraction. If you shade 3 of 6 equal parts, you shade **half** the shape. If you shade 4 of 10 equal parts, you shade **less than half**. So changing the 4 to a 3 makes the right side even smaller. Instead, change **4 to 5**: \[ \frac{3}{6}=\frac{5}{10} \] Both fractions show half of a shape shaded.

Nice attempt — but let's slow down and check your comparison, because there's a tricky idea hiding in it. You said 4/10 is bigger than 3/6 because 4 is bigger than 3 and 10 is bigger than 6. That rule doesn't work for fractions! The denominator tells you the **size of the pieces** — the more pieces you split something into, the smaller each piece gets. Try it with area models: draw two rectangles the same size. Split one into 6 equal parts and shade 3 — you've shaded exactly **half**. Split the other into 10 equal parts and shade 4 — that's **less than half**, because half of 10 parts would be 5 parts. So 4/10 is actually *smaller* than 3/6, not bigger. That's also why your fix, 3/6 = 3/10, doesn't work. Both have 3 shaded pieces, but tenths are smaller pieces than sixths, so 3/10 is less than 3/6. They're not equal. Here's a hint to get you to a true equation. Since 3/6 is exactly one half, you need to turn 4/10 into one half by changing just one number. Two ways to think about it: - Half of 10 pieces is ___ pieces. (This changes the 4.) - 4 shaded pieces would be half of ___ total pieces. (This changes the 10.) Fill in either blank, then draw the area models to check that the two fractions really shade the same amount. Can you find both solutions?

You're on the right track thinking about how the fractions compare, but let's take a closer look at that comparison. You said: *"4/10 is bigger than 3/6, because 4 is bigger than 3 and 10 is bigger than 6."* That reasoning is a common trap! Comparing numerators and denominators separately doesn't always work. Let's use an **area model** (or common denominators) to see which fraction is actually larger. **Let's draw it (or imagine it):** * **3/6**: Draw a rectangle divided into 6 equal parts. Shade 3. That's exactly **half** the rectangle. * **4/10**: Draw a same-sized rectangle divided into 10 equal parts. Shade 4. That's **less than half** (since half would be 5/10). So, **3/6 is actually bigger than 4/10** (3/6 = 5/10, and 5/10 > 4/10). Since 3/6 is the bigger one, you have two main options to make them equal by changing **only one number**: 1. **Make the first fraction smaller** (change the 3 or the 6). 2. **Make the second fraction bigger** (change the 4 or the 10). **Let's test changing the 4 (like you tried), but go the other way:** If we keep the denominator 10, what numerator makes it equal to 3/6 (which is 1/2)? **3/6 = ?/10** → Half of 10 is 5. So **3/6 = 5/10**. (Change the **4 to a 5**). **Or, let's test changing the 10:** If we keep the numerator 4, what denominator makes it equal to 1/2? **3/6 = 4/?** → 4 is half of 8. So **3/6 = 4/8**. (Change the **10 to an 8**). **Your turn:** Can you see why changing the 4 to a **5** works, but changing it to a 3 didn't? Which of the two correct answers (changing 4 to 5, or 10 to 8) makes the most sense to you using the area model?

I like that you started by asking which fraction is bigger. That's a smart first step! But the rule "4 is bigger than 3 and 10 is bigger than 6, so 4/10 is bigger" doesn't work for fractions. Let's check with area models. **Draw two same-size rectangles.** - Cut the first one into **6** equal parts and shade **3**. - Cut the second one into **10** equal parts and shade **4**. ``` 3/6: [■][■][■][ ][ ][ ] 4/10: [■][■][■][■][ ][ ][ ][ ][ ][ ] ``` What do you notice? In the first rectangle, exactly **half** is shaded. In the second one, **less than half** is shaded. So 4/10 is actually *smaller* than 3/6, not bigger. Why? When you cut the same whole into **10** parts instead of **6**, each piece gets **smaller**. So 4 pieces of size "tenths" don't cover as much as 3 pieces of size "sixths." **What happened with your change?** You made it 3/10. Shade 3 out of 10. Is that closer to half, or farther away? Changing the 4 to a 3 moved it the wrong way, so 3/6 = 3/10 isn't true. **Your turn:** 1. 3/6 is exactly half. Look at the rectangle with 10 parts. How many parts would you need to shade to cover exactly half? Change one number to make that true. 2. **Bonus:** There's another way to fix it! Keep the 4 and change the **10** instead. How many equal parts should the rectangle have so that 4 shaded parts make exactly half? Draw it out and tell me what you find!

Let’s compare the shaded fractions using 1/2 as a halfway point. Rectangle R has 4/5 shaded. Since 5/10 is 1/2, 8/10 is more than 1/2. So 4/5 > 1/2. Rectangle S has 3/8 shaded. Since 4/8 is 1/2, 3/8 < 1/2. That means 4/5 > 3/8, so R has more shading than S. Try writing the completed comparison in the blank.

I see why 8 caught your eye—it’s bigger than 5. But the pieces in S are **smaller**, because the same-sized rectangle is cut into more parts. Look at how much of each whole is shaded: \(4/5\) is **more than half**, while \(3/8\) is **less than half**. So R has more shaded, and the correct sign is **\(>\)**: \[ \frac45>\frac38 \]

Good effort jumping in! I can see you're thinking carefully about the two rectangles. But let's look at your reasoning again, because there's a tricky idea hiding here. You said S is bigger because it's cut into 8 parts and R is only cut into 5. It's true that 8 is more than 5 — but here's the catch: **the more equal parts you cut something into, the smaller each part is.** Think about sharing a pizza. Would you get a bigger slice if the pizza were cut into 5 slices or 8 slices? Cutting it into 8 means smaller slices! So instead of counting the parts, let's compare each fraction to a **benchmark** like 1/2: - **Rectangle R: 4/5 shaded.** Half of 5 parts would be 2½ parts. Since 4 is more than 2½, this is *more than half*. In fact, only 1 part out of 5 is unshaded — it's almost the whole rectangle! - **Rectangle S: 3/8 shaded.** Half of 8 parts is exactly 4 parts (4/8 = 1/2). Since only 3 parts are shaded, this is *less than half*. Now put it together: R has **more than half** shaded, and S has **less than half** shaded. So: $$\frac{4}{5} > \frac{3}{8}$$ The correct symbol is **>**. Try sketching the two rectangles and shading them — you'll see right away that R is almost completely filled while S is less than half filled. Does that match what you see?

You're on the right track by noticing the rectangles are cut into different numbers of parts! However, the number of pieces (the denominator) only tells us the *size* of each piece—it doesn't tell us the total shaded amount by itself. We also have to look at how many of those pieces are shaded (the numerator). Let's use **benchmark fractions** (like $\frac{1}{2}$) to help compare: **Rectangle R:** 4 out of 5 parts are shaded ($\frac{4}{5}$). * Think about $\frac{1}{2}$. Half of 5 is 2.5. Since 4 is much more than 2.5, **$\frac{4}{5}$ is greater than $\frac{1}{2}$**. In fact, it's very close to 1 whole rectangle. **Rectangle S:** 3 out of 8 parts are shaded ($\frac{3}{8}$). * Think about $\frac{1}{2}$. Half of 8 is 4. Since 3 is less than 4, **$\frac{3}{8}$ is less than $\frac{1}{2}$**. **Comparison:** * Rectangle R is shaded **more than half**. * Rectangle S is shaded **less than half**. Therefore, the fraction of R shaded is **greater than** the fraction of S shaded. The correct symbol is **>**. **$\frac{4}{5} > \frac{3}{8}$**

You noticed that S is cut into more pieces than R. That's a real difference between the two rectangles. But more pieces doesn't mean more is shaded. Let's look again. **What happens when you cut into more pieces?** Picture sharing one pizza. If you cut it into 8 slices, is each slice bigger or smaller than if you cut it into 5 slices? Each slice gets **smaller**, because the same pizza is split among more pieces. So each piece of S (an eighth) is smaller than each piece of R (a fifth). **Now count how many pieces are shaded.** - R: **4** of its 5 pieces are shaded. Those are the bigger pieces. - S: only **3** of its 8 pieces are shaded. Those are the smaller pieces. **Use the benchmark ½ to check.** - R is cut into 5 parts. Half of 5 is 2½. R has **4** shaded, which is more than half. So 4/5 > ½. In fact, R is almost all shaded. - S is cut into 8 parts. Half of 8 is 4. S has only **3** shaded, which is less than half. So 3/8 < ½. One rectangle is more than half shaded, and the other is less than half shaded. So R has more shaded: **4/5 > 3/8** The correct sign is **>**, not <. **Tip to remember:** The bottom number (denominator) tells you how *big* each piece is. A bigger denominator means *smaller* pieces. Always check how many pieces are shaded too, and comparing each fraction to ½ is a quick way to decide. Want to try one? Which is bigger, **2/6** or **5/7**? Use ½ to help you.

Let’s compare these fractions using equal-sized pieces. We can write them all with the denominator 24: 3/4 = 18/24 1/6 = 4/24 5/8 = 15/24 With the same denominator, we compare the numerators. Since 4 < 15 < 18, the fractions in order from least to greatest are: 1/6, 5/8, 3/4 Try writing these three fractions from greatest to least.

I see why you looked at the denominators, but they don’t tell you the order by themselves: the numerators matter too. Let’s give all three fractions the same denominator so we can compare them: \[ \frac34=\frac{18}{24},\qquad \frac16=\frac{4}{24},\qquad \frac58=\frac{15}{24}. \] Since \(4<15<18\), the correct order from least to greatest is **\(\frac16,\ \frac58,\ \frac34\)**.

Good effort, Devon — but let's look at that rule about denominators, because it actually works the opposite way! Think about a pizza. If you cut it into 4 pieces, each piece is pretty big. If you cut it into 8 pieces, each piece is much smaller. So a **bigger denominator means smaller pieces**, not a smaller fraction. For example, 1/4 is *greater* than 1/8, even though 4 is less than 8. Since these fractions have different numerators too, let's use a handy trick: compare each one to **1/2**. - **1/6**: Half of 6 is 3, so 3/6 = 1/2. Since 1 is less than 3, **1/6 is less than 1/2**. - **5/8**: Half of 8 is 4, so 4/8 = 1/2. Since 5 is more than 4, **5/8 is more than 1/2**. - **3/4**: Half of 4 is 2, so 2/4 = 1/2. Since 3 is more than 2, **3/4 is more than 1/2**. Right away, that tells us 1/6 is the smallest. Now we just compare 5/8 and 3/4. If we rename 3/4 as 6/8, it's easy: 6/8 is greater than 5/8. **Correct order from least to greatest: 1/6, 5/8, 3/4** You can check with decimals too: 1/6 ≈ 0.17, 5/8 = 0.625, and 3/4 = 0.75. Nice work being willing to explain your thinking — that's how mistakes get found!

It looks like you're paying close attention to the denominators, which is a great start! However, the size of a fraction depends on **both** the numerator and the denominator working together. **The mistake:** You ordered them by the denominator only (4, 6, 8). But a smaller denominator doesn't always mean a smaller fraction. For example, $\frac{3}{4}$ is almost a whole, while $\frac{1}{6}$ is a very small piece. Let's check the actual sizes using a common denominator (like 24) or by comparing them to benchmarks like $\frac{1}{2}$: * **$\frac{1}{6}$** $\approx$ 0.16 (Much less than $\frac{1}{2}$) * **$\frac{5}{8}$** = 0.625 (More than $\frac{1}{2}$) * **$\frac{3}{4}$** = 0.75 (More than $\frac{1}{2}$, and bigger than $\frac{5}{8}$) **Correct order from least to greatest:** **$\frac{1}{6}$, $\frac{5}{8}$, $\frac{3}{4}$** **Why?** * $\frac{1}{6}$ is the only one smaller than $\frac{1}{2}$. * Between $\frac{5}{8}$ and $\frac{3}{4}$: $\frac{3}{4}$ is the same as $\frac{6}{8}$, which is bigger than $\frac{5}{8}$. Try drawing three same-sized rectangles and shading in each fraction—it helps to see which shaded area is smallest!

You noticed something real: the denominators do go 4, 6, 8. But ordering by denominator is actually the mistake Devon made, so your order matches his. Let's figure out why it doesn't work. **What does the denominator tell us?** The denominator tells how many equal pieces the whole is cut into. - 1/4 means the whole is cut into 4 pieces. Those are big pieces. - 1/8 means the whole is cut into 8 pieces. Those are smaller pieces. So a bigger denominator means **smaller** pieces, not a bigger fraction. And the denominator is only half the story. The **numerator** tells how many of those pieces you have. **Let's think about each fraction.** - **1/6**: only 1 small piece out of 6. That's not much of the whole. - **5/8**: 5 out of 8 pieces. Half of 8 is 4, so 5/8 is a little **more than half**. - **3/4**: 3 out of 4 pieces. That's more than half too. It's only 1 piece away from a whole. **Try it yourself:** 1. Which fraction is **less than half**? That one should come first. 2. Now compare 5/8 and 3/4. Here's a hint: 3/4 is the same as 6/8. Is 5/8 or 6/8 bigger? Once you've answered those two questions, write the three fractions in order from least to greatest. Then finish this sentence: "Devon's mistake was ________." Send me your new order and I'll check it with you!