Question

math reasoning number sense

1. write the letter that shows where $2\frac{3}{4}$ is located on this number line.

number line with points a (0), b, c, d, e (1), f, g, h, i (2), j, k, l, m (3), n, o, p, q (4)
explain how you decided.
page 165

Explanation:

Step1: Analyze the number line

The number line has marks at 0, 1, 2, 3, 4. Between 2 and 3, we need to find the position of \(2\frac{3}{4}\). First, recall that \(2\frac{3}{4}=\frac{11}{4} = 2.75\). The distance between 2 and 3 is divided into equal parts. Let's see the number of intervals between 2 and 3. From the number line, between 2 (I) and 3 (M), let's count the ticks: I (2), J, K, L, M (3). So there are 4 intervals between 2 and 3 (since from I to M, there are 4 spaces: I to J, J to K, K to L, L to M). Wait, actually, the number of segments between 2 and 3: if M is 3, then from I (2) to M (3), how many steps? Let's check the labels: I is 2, then J, K, L, M is 3. So between 2 and 3, there are 4 equal parts (since 3 - 2 = 1, divided into 4 parts, each part is \(\frac{1}{4}\)). So each tick between 2 and 3 represents an increase of \(\frac{1}{4}\).

Step2: Calculate the position of \(2\frac{3}{4}\)

\(2\frac{3}{4}\) is 2 + \(\frac{3}{4}\). Since each part between 2 and 3 is \(\frac{1}{4}\), we need to move 3 parts from 2 (I). Starting at I (2), first part: J (2 + \(\frac{1}{4}\) = \(2\frac{1}{4}\)), second part: K (2 + \(\frac{2}{4}\) = \(2\frac{2}{4}\) = \(2\frac{1}{2}\)), third part: L (2 + \(\frac{3}{4}\) = \(2\frac{3}{4}\))? Wait, no, wait the labels: I is 2, then J, K, L, M is 3. Wait, maybe I made a mistake. Let's re - examine the number line:

Labels: A(0), B, C, D, E, F, G, H, I(2), J, K, L, M(3), N, O, P, Q(4)

So between I (2) and M (3), the labels are I, J, K, L, M. So the number of intervals between 2 and 3 is 4 (because from I to M, there are 4 gaps: I - J, J - K, K - L, L - M). So each interval has a length of \(\frac{3 - 2}{4}=\frac{1}{4}\). So to find \(2\frac{3}{4}\), we start at 2 (I) and move 3 intervals (since \(\frac{3}{4}\) is 3 times \(\frac{1}{4}\)). So from I (2), first interval: J (\(2+\frac{1}{4}=2\frac{1}{4}\)), second: K (\(2+\frac{2}{4}=2\frac{1}{2}\)), third: L (\(2+\frac{3}{4}=2\frac{3}{4}\))? Wait, but M is 3, so L is before M. Wait, \(2\frac{3}{4}\) is 2.75, and 3 is 3.0. So between 2 and 3, each \(\frac{1}{4}\) increment: 2.0 (I), 2.25 (J), 2.5 (K), 2.75 (L), 3.0 (M). Yes! So \(2\frac{3}{4}=2.75\), which is at L. Wait, but let's confirm:

\(2\frac{3}{4}\) is equal to 2 + \(\frac{3}{4}\). Since the distance from 2 to 3 is 1 unit, and it's divided into 4 equal parts (because from I to M, there are 4 intervals: I - J, J - K, K - L, L - M), each part is \(\frac{1}{4}\). So moving 3 parts from 2 (I) gives \(2+\frac{3}{4}=2\frac{3}{4}\), which is at the label L? Wait, no, wait the labels: I is 2, then J, K, L, M is 3. So I (2), J (2.25), K (2.5), L (2.75), M (3). Yes, so \(2\frac{3}{4}\) is at L? Wait, but let's check the original number line again. Wait, maybe I miscounted the intervals. Wait, the problem says "Write the letter that shows where \(2\frac{3}{4}\) is located". Let's re - express \(2\frac{3}{4}\): it's 2 and 3/4. So between 2 and 3, we need to find the 3/4 mark. Since 1 (from 2 to 3) is divided into 4 parts (because from I to M, there are 4 spaces: I to J, J to K, K to L, L to M), each part is 1/4. So 3/4 is 3 of those parts. So starting at I (2), first part: J (2 + 1/4), second: K (2 + 2/4), third: L (2 + 3/4), fourth: M (2 + 4/4 = 3). So \(2\frac{3}{4}\) is at L? Wait, but maybe the number of intervals is different. Wait, let's count the ticks between 2 and 3: I is 2, then J, K, L, M is 3. So that's 4 intervals (between I and J, J and K, K and L, L and M), so 4 equal parts, each 1/4. So 2 + 3/4 is at L. Wait, but let's confirm with the fraction: \(2\frac{3}{4}=\frac{11}{4}\), and the number line from 0 to 4, with marks at 0,1,2,3,4. Between 2 and 3, the position of \(2\frac{3}{4}\) is 3/4 of the way from 2 to 3. Since the segment from 2 to 3 is divided into 4 equal parts (as per the number of letters between I and M: I, J, K, L, M – so 4 gaps), each part is 1/4. So 3 parts from 2 is 3/4, so the letter is L? Wait, but maybe I made a mistake. Wait, let's check the answer again. Wait, maybe the number of intervals between 2 and 3 is 4, so each interval is 1/4. So 2 + 3/4 is at L. So the letter is L? Wait, but let's see the original problem's number line: A(0), B, C, D, E, F, G, H, I(2), J, K, L, M(3), N, O, P, Q(4). So between I (2) and M (3), the letters are J, K, L. Wait, I to J is 1st, J to K 2nd, K to L 3rd, L to M 4th. So 2 + 3/4 is at L. So the answer should be L.

Answer:

The letter is L. The explanation is: \(2\frac{3}{4}\) is \(2+\frac{3}{4}\). The interval between 2 (I) and 3 (M) is divided into 4 equal parts (since from I to M there are 4 intervals: I - J, J - K, K - L, L - M), each part is \(\frac{1}{4}\). Moving 3 parts from 2 (I) gives \(2+\frac{3}{4}=2\frac{3}{4}\), which corresponds to the letter L.