Operations with rational numbers
Adding, subtracting, multiplying and dividing positive and negative rational numbers, and solving real-world problems with them
About four lessons of 45 to 60 minutes
A submarine, a bank balance and a thermometer all use the same math
A submarine's depth, a bank balance that can go into overdraft, and a temperature that can drop below zero all use positive and negative numbers, together called rational numbers when fractions and decimals are included. The rules for combining them, adding, subtracting, multiplying and dividing, are exactly the same whether the number is -8, -3/4, or -0.6.
Once those rules are automatic, real problems become a matter of translation: 'rises then drops' is addition then subtraction; 'a rate for several minutes' is multiplication. This unit locks in the sign rules and then puts them to work on real multistep problems.
- A temperature that rises then drops overnightadding a positive change, then a negative one
- A diver descending at a steady rate for several minutesa negative rate multiplied by a positive time
- A bank balance in overdraft (negative)spending and earning both move the balance along the number line
- Elevation above and below sea levelclimbing and descending combine with addition and subtraction of signed numbers
What students will be able to do
Students will add and subtract rational numbers (including negatives) and locate the results on a number line, multiply and divide rational numbers by correctly applying the rules for signs, and solve multistep real-world problems that combine all four operations with positive and negative rational numbers.
- I can add and subtract positive and negative rational numbers, including fractions and decimals.
- I can show an addition or subtraction of rational numbers as a jump on a number line.
- I can multiply and divide rational numbers, correctly applying the rule for signs (same signs give a positive result, different signs give a negative result).
- I can solve a real-world multistep problem that uses more than one operation with rational numbers.
Standards this unit teaches
- 7.NS.A.1Common Core (US)Add and subtract rational numbers
Add and subtract rational numbers, including negatives, and understand them on a number line.
- 7.NS.A.2Common Core (US)Multiply and divide rational numbers
Multiply and divide rational numbers, applying the rules for signs and properties of operations.
- 7.NS.A.3Common Core (US)Solve problems with rational numbers
Solve real world problems using all four operations with rational numbers.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Grade 6 integers & negative numbers teaching unitthe number-line model and comparing negatives this unit builds directly on
- Grade 6 dividing fractions teaching unitdividing by a fraction (multiply by the reciprocal) still applies once negatives are involved
- Integer in the glossarya refresher on whole numbers and their negatives, before fractions and decimals join them
- Negative number in the glossarya quick refresher on what a negative number represents
Words to teach and display
- Rational number
- any number that can be written as a fraction of two integers, including negative fractions and decimals
- Opposite (additive inverse)
- a number the same distance from 0 but on the other side, such as -5 being the opposite of 5
- Absolute value
- a number's distance from 0 on the number line, always zero or positive
- Reciprocal
- the number you multiply by to get 1; dividing by a fraction is the same as multiplying by its reciprocal
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Adding and subtracting rational numbers
ConcreteAdding a positive number moves right on the number line; adding a negative number (or subtracting a positive one) moves left. Subtracting a negative number is the trickiest case: it moves RIGHT, because subtracting a negative is the same as adding its opposite.
Think of temperature: -8°C rises 15°C to reach 7°C (a move right of 15), then drops 20°C overnight to reach -13°C (a move left of 20). Every addition or subtraction of a rational number is just a jump of that size, in that direction, from the starting point.
The subtracting-a-negative rule follows the same idea: 5 - (-3) does not mean 'go left 3 from 5'. It means 'take away a debt of 3', which actually makes you 3 richer: 5 - (-3) = 5 + 3 = 8.
The temperature is -8°C. It rises 15°C during the day, then drops 20°C overnight. Find the overnight temperature.
- Rise: -8 + 15 = 7.
- Drop: 7 - 20 = -13.
Answer: The overnight temperature is -13°C.
- Why does 5 - (-3) give a bigger number than 5, not a smaller one?
- When adding two numbers with different signs, how do you find the sign of the answer?
2. Multiplying and dividing rational numbers
PictorialMultiplying and dividing rational numbers uses one consistent sign rule: same signs give a positive result, different signs give a negative result. This applies to fractions and decimals exactly as it does to whole numbers.
For -3/4 x (-2/5): multiply the numerators and denominators as usual, (3 x 2)/(4 x 5) = 6/20 = 3/10, and since both factors were negative (same sign), the result is positive: 3/10.
For -5.6 / 0.8: divide the magnitudes, 5.6 / 0.8 = 7, and since the signs are different (negative divided by positive), the result is negative: -7.
Calculate -3/4 x (-2/5), then -5.6 / 0.8.
- -3/4 x (-2/5): multiply magnitudes, (3 x 2)/(4 x 5) = 6/20 = 3/10. Two negatives multiplied give a positive result.
- -5.6 / 0.8: divide magnitudes, 5.6 / 0.8 = 7. A negative divided by a positive gives a negative result.
Answer: -3/4 x (-2/5) = 3/10, and -5.6 / 0.8 = -7.
- If you multiply three negative numbers together, is the result positive or negative? Why?
- Does the sign rule for dividing rational numbers work any differently from the rule for multiplying?
3. Solving real-world problems with all four operations
AbstractReal problems rarely announce which operation to use, so translate the situation first: a repeated rate over time is multiplication, a one-time change is addition or subtraction, and the final answer combines them in the order the story happens.
A scuba diver starts at a depth of -12 meters. She descends at a rate of -3 meters per minute for 4 minutes (a repeated rate, so multiply: -3 x 4 = -12, the total change from descending), then ascends 10 meters (a one-time change, add +10). Combine every change with the starting depth in order.
A diver starts at -12 meters. She descends at -3 meters per minute for 4 minutes, then ascends 10 meters. Find her final depth.
- Total change from descending: rate x time = -3 x 4 = -12.
- Apply it to the starting depth: -12 + (-12) = -24.
- Apply the ascent: -24 + 10 = -14.
Answer: Her final depth is -14 meters (14 meters below the surface).
- Why is the descent modeled as a multiplication (rate x time) rather than just another addition?
- What would change in the calculation if she had ascended 10 meters BEFORE descending, instead of after?
Common misconceptions and how to address them
MisconceptionSubtracting a negative number gives a smaller (more negative) result, e.g. 5 - (-3) = 2.
Why it happens: The two negative signs are read as 'even more negative' instead of recognizing that subtracting a negative reverses to addition.
How to address it: Rewrite every 'subtract a negative' as 'add its opposite' before computing: 5 - (-3) becomes 5 + 3 = 8. Practice the rewrite as a standalone step until it is automatic.
MisconceptionA negative number times a negative number is negative, because 'negative always wins'.
Why it happens: Students overgeneralize that negatives dominate any calculation they appear in.
How to address it: Two negatives multiplied give a POSITIVE result. Count the negative factors in a multiplication: an even count gives a positive result, an odd count gives a negative result.
MisconceptionWhen adding two numbers with different signs, subtract their magnitudes but keep the sign of the FIRST number written, regardless of which magnitude is bigger.
Why it happens: Students apply a left-to-right sign rule instead of comparing which number's magnitude is actually larger.
How to address it: Subtract the smaller magnitude from the larger one, and give the result the sign of the number with the LARGER magnitude, not the first number written. -8 + 15: 15 has the larger magnitude, so the answer is positive, 7.
MisconceptionDivision does not follow a sign rule the way multiplication does, so a negative divided by a positive can come out positive.
Why it happens: Students learn the multiplication sign rule well but do not realize division follows the identical rule.
How to address it: Division and multiplication use the SAME sign rule: same signs give a positive result, different signs give a negative result. -5.6 / 0.8 is negative because the signs differ.
MisconceptionIn a multistep real-world problem, every number is just added or subtracted directly, skipping any multiplication a repeated rate needs.
Why it happens: Students see two given numbers (a rate and a time) and default to combining them the same way as the other numbers in the problem, with addition.
How to address it: Before combining anything, ask: is this a ONE-TIME change (add or subtract it) or a RATE repeated over time (multiply rate by time first, then add the result)? Identify the operation the situation models before computing.
Guided practice (with answers)
1. Calculate: -6 + 9
Answer: 3
2. Calculate: 4 - (-7)
Answer: 11, because 4 - (-7) = 4 + 7 = 11.
3. Calculate: -3.5 x 4
Answer: -14, because the signs are different (negative x positive), so the result is negative.
4. Calculate: -2/3 x (-3/4)
Answer: 1/2, because (2 x 3)/(3 x 4) = 6/12 = 1/2, and two negatives give a positive result.
5. Calculate: -18 / (-6)
Answer: 3, because a negative divided by a negative (same sign) gives a positive result.
6. A hiker starts at an elevation of -25 feet and climbs 8 feet per hour for 3 hours. Find the elevation after 3 hours.
Answer: -1 foot, because the climb is 8 x 3 = 24 feet, and -25 + 24 = -1.
Independent practice worksheets
Practise the four operations with signed rational numbers, and multistep real-world problems, with computed, never-wrong answer keys.
Differentiation
- Model every addition and subtraction with an actual number-line jump before moving to a rule stated in words.
- Practice 'subtract a negative = add its opposite' as its own rewriting drill, separate from computing the answer, until the rewrite is automatic.
- Use a simple sign-counting table for multiplication and division: count negative factors, even is positive, odd is negative.
- For multistep word problems, underline the rate and the time separately before writing any calculation, so the multiplication step is not skipped.
- Introduce problems mixing fractions, decimals and negatives in the same calculation, e.g. -2 1/2 x 0.4.
- Ask students to write their own real-world story problem that requires all four operations with rational numbers, then solve it.
- Explore why the product of an even number of negative factors is always positive, using a general argument rather than just examples.
- Investigate multistep problems where the order of the one-time changes affects an intermediate value but not the final answer (addition is commutative), versus ones where order genuinely matters (e.g. percent changes, from the ratios unit).
Assessment: exit ticket
A three-question exit ticket sampling addition/subtraction, multiplication/division, and a multistep real-world problem.
1. Calculate: -9 + (-4)
Answer: -13
2. Calculate: -2/5 x 10
Answer: -4, because 10 x (-2/5) = -4 (different signs give a negative result).
3. A submarine at -150 feet dives another 45 feet, then rises 60 feet. Find its final depth.
Answer: -135 feet, because -150 - 45 = -195, and -195 + 60 = -135.
Teacher notes and timings
- Rough timing across four lessons: Lesson 1 adding and subtracting (section 1), Lesson 2 multiplying and dividing (section 2), Lesson 3 multistep real-world problems (section 3), Lesson 4 mixed review plus the exit ticket.
- This unit assumes comfort with integers on a number line and dividing fractions (Grade 6 units). Revisit those first if either foundation is shaky before adding negatives to the mix.
- Language to keep repeating: 'subtract a negative, add its opposite'; 'same signs positive, different signs negative' (for both multiplying and dividing); 'a repeated rate is multiplied, a one-time change is added or subtracted'.
- The number-line figure in section 1 deliberately narrates both jumps (rise, then drop) so students see subtraction and addition as the same kind of move, just in opposite directions.
- Curriculum note: 7.NS.A.1-3 (Common Core) are usually taught as one connected sequence, since 7.NS.A.3's real-world problems require the sign rules from 7.NS.A.1 and 7.NS.A.2 to already be automatic.
- Present mode and print both work: use Present to narrate the number-line jumps live with the class, then print the worksheets for independent practice.