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# Operations on the number line

## Task

A number line is shown below. The numbers $0$ and $1$ are marked on the line, as are two other numbers $a$ and $b$.

Which of the following numbers is negative? Choose all that apply. Explain your reasoning.

- $a - 1$
- $a - 2$
- $-b$
- $a + b$
- $a - b$
- $ab + 1$

## IM Commentary

There is a distinction in the Common Core State Standards between a * fraction* and a * rational number*. Fractions are always positive, and when thinking of the symbol $\frac{a}{b}$ as a fraction, it is possible to interpret it as $a$ equal-sized pieces where $b$ pieces make one whole. The rational numbers are the set of fractions taken together with their opposites: understanding rational numbers requires understanding both fractions and signed numbers. The standard 7.NS.1 signals a significant shift from working exclusively with positive numbers to working with signed numbers. The focus of this task is on the nature of signed numbers rather than the "part-whole" interpretation of fractions.

The purpose of this task is to help solidify students' understanding of signed numbers as points on a number line and to understand the geometric interpretation of adding and subtracting signed numbers. This task (like all tasks featured on the Illustrative Mathematics website) assumes that the number line is drawn to scale.

## Solution

- $a$ is greater than $1$, so $a - 1$ is positive.

- The distance between $a$ and $1$ appears to be less than the distance between $1$ and $0$, so it looks like $a$ is less than $2$. Thus $a-2$ is negative.

- $b$ is negative, so $-b$ is positive.

- The distance between $a$ and $0$ appears to be less than the distance between $b$ and $0$, so it looks like $|a|$ is less than $|b|$. Since $b$ is negative and a is positive, $a+b$ is negative.

- $a-b$ = $a+ -b$. Since $b$ is negative, $-b$ is positive. $a$ is also positive. Thus, $a-b$ is positive.

- Since $|a|$ and $|b|$ are both greater than $1$, $|ab|$ is also greater than 1 (this builds on the intuition students gained in fifth grade as in 5.NF.5). $ab$ is negative since $a$ is positive and $b$ is negative. Thus, $ab+1$ is negative.

## Operations on the number line

A number line is shown below. The numbers $0$ and $1$ are marked on the line, as are two other numbers $a$ and $b$.

Which of the following numbers is negative? Choose all that apply. Explain your reasoning.

- $a - 1$
- $a - 2$
- $-b$
- $a + b$
- $a - b$
- $ab + 1$

## Comments

Log in to comment## Ashli says:

about 5 yearsI'm curious how revisiting a task like this a week later but with students marking off their own a, b, and 1 values would go. perhaps give blank number lines then trade and answer the same questions as the ones here. Perhaps as they develop more number sense turn the question around and have them place down a, b, 1 so that specific combinations are negative while others are positive. Brownie points to any teacher that makes this happen and reports back :)

## smckj says:

over 6 yearsI think this question should state that the comparative distances are accurate. Otherwise none of the solutions could be proven to be negative. If CCSS are supposed to help students think mathematically, they should reflect mathematical ideas including the concept that one can't make decisions(e.g., if an angle is 90 degrees) based on how something appears.

## Emily Abshier says:

about 5 yearsThanks for the excellent thoughts. Because if this, I was prepared for those conversations when I did this in class. I went ahead and told students they could assume distances were what they looked like, but also that any student who felt that he was advanced, and wanted a 4 (rubric-grading) needed to explain any such assumption he made. It allowed advanced students to show me so much more of what they knew while other students could still master the task. An advanced student might say, "negative as long as m is longer than n." Another student might just assume that fact and draw an arrow of length m attached to n.

## Cam says:

over 6 yearsYou bring up a good point, and one that has spawned many discussions of this type (hence, for example, the site-wide convention that number lines are drawn to scale). It does seem clear that sensory observation need not be summarily dismissed in geometry problems, especially in a setting prior to the introduction of formal proofs in 8th grade. To take it to an extreme, do we also question whether 1 is really between 0 and a, just because it appears that way in the diagram? There's also, to me, a significant difference between eye-balling an open condition ("that distance is greater than that other distance"), and a closed condition ("that angle measurement is precisely 90 degrees").

In any case, this assumption is made explicit in the commentary, so a teacher who felt there was a benefit of mentiong it, or room for confusion without, could either include this verbally or begin a discussion on it.