Arithematics
Differences, halfway points and number lines
In this note
- 1.6Find the difference between two numbers, including negatives (temperatures, heights).
- 1.6Find the number halfway between two numbers, fractions or decimals, and a missing end.
- 1.6Find values at equally spaced points on a number line.
Before you start: negative numbers and fractions.
Formula bank
The difference between two numbers
General methodHow to find any difference
- Decide which number is bigger (on a number line, further right).
- Bigger − smaller. With a negative this is subtracting a negative: 18 − (−3) = 21.
- The answer is positive. "9 °C below −4 °C" is a different question: −4 − 9 = −13.
Example Inside 21 °C, outside −6 °C. Find the difference.
Bigger − smaller.
- \(21-(-6)=21+6\)
Answer 27 °C
Try it yourself
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Answer 20 °C
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Answer 8 °C
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Answer −13 °C
The number halfway between
General methodHow to find the number halfway between two numbers
- Add the two numbers (fractions: common denominator first).
- Halve the total.
- Missing end: double the middle number and take away the end you know.
Why: halfway is the start plus half the gap, \(a+\tfrac{b-a}2=\tfrac{a+b}2\).
Example Find the fraction halfway between \(\tfrac23\) and \(\tfrac34\).
Add with a common denominator, then halve.
- \(\tfrac8{12}+\tfrac9{12}=\tfrac{17}{12}\)
- \(\tfrac{17}{12}\div2\)
Answer \(\tfrac{17}{24}\)
Try it yourself
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Answer 2
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Answer \(\tfrac5{12}\)
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Answer 6
Equal steps on a number line
General methodHow to find marks on a number line split into equal parts
- Gap = end − start.
- One step = gap ÷ number of equal parts.
- The kth mark = start + k × step.
Example Find the number one third of the way from \(\tfrac14\) to \(\tfrac12\).
Gap, then a third of it, added to the start.
- gap \(=\tfrac14\)
- a third \(=\tfrac1{12}\)
- \(\tfrac14+\tfrac1{12}=\tfrac4{12}\)
Answer \(\tfrac13\)
Try it yourself
Hint
Show answer
Answer 5
Ways they test it
Type 1 Temperature difference
"Inside 21 °C, outside −6 °C. Find the difference." (1 mark)
Spot it two temperatures.
- Bigger − smaller.
Type 2 A table of temperatures
"Find the greatest increase from one month to the next." (2 marks)
Spot it a table of values.
- Each neighbouring change.
- Pick the largest increase.
For example …, −3, 4, 11, …
- −3 → 4: +7
- 4 → 11: +7
Answer 7 °C
Type 3 Halfway between fractions
"Find the fraction halfway between \(\tfrac23\) and \(\tfrac34\)." (2 marks)
Spot it "halfway" with fractions.
- Add (common denominator).
- Halve.
Type 4 Equal steps
"P and Q split the line from \(\tfrac15\) to \(\tfrac12\) into three equal parts. Find P and Q."
Spot it "equal parts".
- Gap ÷ parts.
- Step along from the start.
For example \(\tfrac15\) to \(\tfrac12\), 3 parts
- gap \(\tfrac3{10}\), step \(\tfrac1{10}\)
Answer P \(=\tfrac3{10}\), Q \(=\tfrac25\)
Mistakes to avoid
- ✗ 21 − 6 = 15✓ 21 − (−6) = 27Subtract the negative.
- ✗ Halfway between \(\tfrac23\) and \(\tfrac34\) = \(\tfrac57\)✓ \(\tfrac{17}{24}\)Never add tops and bottoms.
- ✗ Halfway = difference ÷ 2✓ start + difference ÷ 2Half the gap is added to the start.
- ✗ A negative difference✓ always positiveIt is a distance.
Summary
- Difference = bigger − smaller (positive).
- Halfway = (a + b) ÷ 2.
- Missing end = 2 × middle − known end.
- n equal parts: step = gap ÷ n.
If you see …
- "difference" with a negativesubtract the negative: add.
- "halfway" with fractionscommon denominator, add, halve.
Practice
Practice: easy, medium, hard
Easy
Show working
- \(12-(-8)\)
Answer 20 °C
Show working
- \(0.65\div2\)
Answer 0.325
Show working
- \(-6\div2\)
Answer −3
Medium
Guide me
What is it asking? the halfway fraction
What do we know? denominators 8 and 6
Plan over 24, add, halve
Show working
- \(\tfrac9{24}+\tfrac{20}{24}=\tfrac{29}{24}\)
- \(\div2\)
Answer \(\tfrac{29}{48}\)
Guide me
What is it asking? the difference
What do we know? highest 19, lowest −14
Plan bigger − smaller
Show working
- \(19-(-14)\)
Answer 33 °C
Guide me
What is it asking? the missing end
What do we know? middle 2
Plan 2 × 2 − (−5)
Show working
- \(4+5=9\)
Answer 9
Hard
Guide me
What is it asking? P
What do we know? gap \(\tfrac3{10}\)
Plan one step = gap ÷ 3
Show working
- step \(=\tfrac1{10}\)
- \(\tfrac2{10}+\tfrac1{10}\)
Answer \(\tfrac3{10}\)
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| step \(\tfrac1{10}\) | M1 | ||
| \(\tfrac3{10}\) | A1 | ||
| Total: 2 marks | |||
Guide me
What is it asking? the third mark
What do we know? gap 12, 4 parts
Plan step 3, three steps from −4
Show working
- \(-4+3\times3=5\)
Answer 5
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| step 3 | M1 | ||
| 5 | A1 | ||
| Total: 2 marks | |||
Guide me
What is it asking? the biggest rise
What do we know? neighbouring values
Plan each change, then the largest
Show working
- +7, +7, +5, +3, −2, −7
Answer 7
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| 7 | B1 | ||
| Total: 1 mark | |||
Can I do this?
- I can find the difference between two numbers, including negatives.
- I can find the number halfway between two numbers or fractions, and a missing end.
- I can find values at equal steps on a number line.
Anything unticked: revise that section first. All ticked: take the quiz.
Quiz
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