Math / Grade 6
Integers: position is not distance
Learning goal: Compare signed numbers, locate opposites and explain absolute value with a library lift and equal-interval number lines.
Before you start: Locate zero and positive whole numbers on an equal-interval number line; read less-than and greater-than comparisons.
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Integer Positions: Follow the Number Line
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1. The lift has a basement
Video: 0:00
Our imaginary library has a lift with basement levels. The lift announces level negative five. It has not broken mathematics. We need numbers below our reference level, zero. Integers include zero, the positive whole numbers, and their negatives. On a horizontal number line, negative numbers are left of zero and positive numbers are right of zero. Keep every unit interval the same size. Zero is neither positive nor negative. We are using a pretend lift, so please leave real lift buttons alone. The librarian has enough surprises today.
2. Right means greater
Video: 0:43
Locate negative five and negative one. Negative one is farther right, so negative one is greater. Negative five is less than negative one. The smaller opening of the less than sign points toward the smaller value. Do not compare only the digits five and one and forget the negative signs. A level one floor below zero is higher than a level five floors below zero. Try tracing from left to right: negative five, negative four, negative three, negative two, negative one, zero. Values increase as you move right.
3. Distance answers another question
Video: 1:24
Absolute value asks how far a number is from zero. The two upright bars around a number mean absolute value. Negative five is five units from zero, so its absolute value is five. Distance cannot be negative. Opposites are the same distance from zero on different sides: negative four and positive four are opposites. They are not equal, but their absolute values are equal. Notice the difference: which position is greater and which distance is greater are different questions. Count intervals between marks, not the marks themselves, to measure a distance.
4. Pause: two library labels
Video: 2:07
Pause the video and investigate two new library levels: negative seven and negative three. Which level is greater? Which level is farther from zero? Explain each answer using the number line. Then name the opposite of negative seven. Start by finding zero and checking that the tick intervals are equal. You can point to the line, draw your own, or explain aloud. If the same number wins both comparisons, look carefully at what each question asks. Continue after you have two comparisons and a reason for each one.
5. Check the labels, not the size of a digit
Video: 2:48
Negative three is greater than negative seven because it lies farther right. Negative seven is farther from zero: its distance is seven, while the other distance is three. The opposite of negative seven is positive seven. The signs describe positions; absolute values describe distances from zero. Neither answer needs a guess about which digit looks bigger. A useful check is to read your answer as a sentence. Negative three is a higher level, but negative seven is farther below the reference level. Our library map now has labels that actually help the lift operator.
6. Zero has an unusual mirror image
Video: 3:32
Fun fact: zero is the only number that is its own opposite. Reflect zero across zero and it stays exactly where it is. Reflect any nonzero number and its opposite is on the other side. Continue to the first worksheet to compare positions, distances, and opposites. Then try the second form with different numbers and situations. Explain at least one answer without copying the worked example. A correct choice is useful, but explaining whether you compared position or distance tells you much more about your understanding. The next lesson will move along the line using signed changes.
Show your understanding
You can point, explain aloud, draw or write.
- Order signed integers by position and distinguish that order from their distances to zero.
- Identify opposites and explain absolute value, including zero, using a fresh number-line example.
Try it yourself
Pause to compare -7 and -3 twice: once by position and once by distance from zero. Explain each answer.
Continue to the two worksheets. Use changed numbers and distinguish signed order from absolute value.
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Integer Positions: Follow the Number Linehttps://s3u.com/mi6p1
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