Below zero, then back up. Fractions and decimals are the same numbers in different clothes.
A rational number can be written as a fraction a/b. Here a and b are integers: whole numbers such as 0, 1, 2 and their negatives, such as −1 and −2. The denominator, b, cannot be zero. A terminating decimal stops, like 0.25. A repeating decimal repeats a pattern forever, like 0.333… Both are rational: 0.25 = 1/4 and 0.333… = 1/3.
These number groups fit inside one another. Counting numbers are 1, 2, 3, and so on. Whole numbers also include 0. Integers include whole numbers and their negatives. Every integer is rational: for example, −3 = −3/1. The number 1/2 is rational but is not an integer. π is irrational: it cannot be written as a fraction of two integers.
Signs have a small trick. Adding a negative is the same as subtracting. Subtracting a negative is the same as adding. Two negatives in a multiply or a divide make a positive. One negative stays negative.
To add fractions, first give them the same denominator—the bottom number that tells the size of each piece. For example, 1/2 + 1/4 = 2/4 + 1/4 = 3/4. To divide by a nonzero fraction, swap that second fraction’s top and bottom numbers and multiply; this swapped fraction is its reciprocal. For decimal addition, line up the decimal points so the same place values add together.
Try it. Then open the steps.
Example 1 of 6
A thermometer reads 3°F. It drops 11 degrees, then rises 5. What does it read?
Answer: −3°F
Example 2 of 6
Compute (−2/3) × (9/4).
Answer: −3/2
Example 3 of 6
−4.8 ÷ 0.6
Answer: −8
Example 4 of 6
Dawn, a painted line on the path
You start at −40 feet (below a ridge). You climb 18 feet, then drop 7. Where are you now?
Answer: −29 feet
Example 5 of 6
Friday lights
A team gains 12 yards, loses 8 yards, then gains 5 yards. What is the net change—the total change after all three plays—in yards?
Answer: 9 yards
Example 6 of 6
A nested diagram in a notebook
Is −3 a whole number, an integer, a rational number, or more than one of those?
Answer: integer and rational
Easy to miss: 2 − (−5) looks like it should be −3. Two minuses in a row are not extra negative.
Do this: 2 − (−5) = 2 + 5 = 7. Say it out loud: minus a negative means plus.
Easy to miss: Adding 1/2 + 1/3 as 2/5. That adds the tops and the bottoms.
Do this: Find a common denominator first. 3/6 + 2/6 = 5/6. Never add the bottoms.
29 problems · Camp, Trail, Summit
