Dr J Frost (jfrost@tiffin.kingston.sch.uk) Year 7 Negative Numbers Dr J Frost (jfrost@tiffin.kingston.sch.uk) Objectives: Add, subtract, multiply and divide negative numbers, as well as raise a negative number to a power. Last modified: 18th July 2015

For Teacher Use: Recommended lesson structure: Lesson 1: Adding, subtracting negative numbers. Lesson 2: Multiplying, dividing and mixed questions.

Visualising negatives with a number line -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 For any addition/subtraction of numbers, you should ALWAYS visualise a number line. How would we visualise the following? Click > Click > 3−7=−𝟒 −6+10=𝟒 Click > Click > −7+5=−𝟐 −3−4=−𝟕 Bro Tip: Notice that the answer is always either the sum of the two numbers (ignoring sign) or the difference.

Test Your Understanding Copy and complete the following. 6−11=−𝟓 3−10=−𝟕 −1−5=−𝟔 −10+11=𝟏 −5+1=−𝟒 −13+6=−𝟕 −4−25=−𝟐𝟗 −25+4=−𝟐𝟏 [JMC 2014 Q7] What is 2014−4102? 𝟒𝟏𝟎𝟐−𝟐𝟎𝟏𝟒=𝟐𝟎𝟖𝟖 ∴𝟐𝟎𝟏𝟒−𝟒𝟏𝟎𝟐=−𝟐𝟎𝟖𝟖 a ? b ? c ? d ? e ? f ? g ? h ? N ?

What is the answer. How could we show it? Adding/subtracting negative numbers 9 + −3= ? What is the answer. How could we show it? 9+2=𝟏𝟏 9+1=𝟏𝟎 9+0=𝟗 9+ −1 =𝟖 9+ −2 =𝟕 9+ −3 =𝟔 ? ? ?

What is the answer. How could we show it? Adding/subtracting negative numbers 5 − −2= ? What is the answer. How could we show it? 5 − 2=𝟑 5 − 1=𝟒 5 − 0=𝟓 5 − −1 =𝟔 5 − −2 =𝟕 5 − −3 =𝟖 ? ? ? ! +− ⇒ − A plus and a minus become a minus −− ⇒ + Two minuses become a plus.

“Two minuses make a plus!” Common Misunderstanding: “Two minuses make a plus!” −4−10=14 What did they do wrong? Two minuses only become a plus when they’re next to each other. If we think about a number line can see the above doesn’t make sense. ? Quickfire Questions: Check Your Understanding: 7 + −5 =𝟐 7 − −5 =𝟏𝟐 −7 + −5 =−𝟏𝟐 −7 − −5 =−𝟐 7 + 5 =𝟏𝟐 7 − 5 =𝟐 −7 + 5 =−𝟐 −7 − 5 =−𝟏𝟐 4 + 10 =𝟏𝟒 4 − 10 =−𝟔 −4 + 10 =𝟔 −4 − 10 =−𝟏𝟒 4 + −10 =−𝟔 4 − −10 =𝟏𝟒 −4 + −10 =−𝟏𝟒 −4 − −10 =𝟔 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

Exercise 1 Find two numbers which have a sum of -2 and a difference of 8. Solution: 3 and -5 [JMC 2010 Q1] What is 2010+ +2010 + −2010 − +2010 −(−2010) ? Solution: 2010 [JMC 2013 Q6] What is the value of 1−1 −1 − 1− 1−1 ? Solution: -2 [JMO 2007 A1] What is the value of 1 5 − 2 4 + 3 3 − 4 2 + 5 1 ? Solution: 1 [IMC 2001 Q1] Between which of the following pairs of numbers is there the greatest difference? A -3, 8 B -5, -13 C 1, 11 D 4, -5 E -6, -15 Solution: A [SMC 2009 Q3] What is the value of 1 6 − 2 5 + 3 4 − 4 3 + 5 2 − 6 1 Solution: 5 [Kangaroo Grey 2004 Q5] What is the value of the expression 1−2 − 3−4 − 5−6 −…− 99−100 Solution: 48 1 Calculate the following. −6+15=𝟗 7−15=−𝟖 4− −7 =𝟏𝟏 9+ −10 =−𝟏 −18+7=−𝟏𝟏 −5− −2 =−𝟑 −20− −13 =−𝟕 −1− −1 =𝟎 −6+ −8 =−𝟏𝟒 −2−(−6)=𝟒 5−(−20)=𝟐𝟓 3+ −24 −4=−𝟐𝟓 1−6− −4 =−𝟏 − 6+ −7 +8=−𝟓 Fill in the missing value. −3+ ∎ =7 ∎=10 7−∎=−10 ∎=17 8+∎=6 ∎=−2 −2−∎=−8 ∎=6 −3−∎=10 ∎=−13 −10+∎=−30 ∎=−20 3 a ? ? b ? 4 c ? d ? e ? ? f ? 5 g ? ? h ? i ? j ? 6 ? k ? l ? m ? 7 n ? ? 2 a ? 8 ? ? b c ? d ? N e ? f ? ?

9× −3 = ? Multiplying Negative Numbers What is the answer? Could we show it in a similar way to how we showed 9+ −3 =6? 9+2=𝟏𝟏 9+1=𝟏𝟎 9+0=𝟗 9+ −1 =𝟖 9+ −2 =𝟕 9+ −3 =𝟔 9×2=𝟏𝟖 9×1=𝟗 9×0=𝟎 9× −1 =−𝟗 9× −2 =−𝟏𝟖 9× −3 =−𝟐𝟕 ? ? ?

−2 × −3 = ? Multiplying Negative Numbers What is the answer? Could we show how to get it by starting with 2× −3 =−6? 2×(−3)=𝟔 1×(−3)=𝟑 0×(−3)=𝟎 (−1)× −3 =−𝟗 (−2)× −3 =−𝟏𝟖 ? ? ! 𝑝𝑜𝑠×𝑛𝑒𝑔=𝑛𝑒𝑔 𝑛𝑒𝑔×𝑝𝑜𝑠=𝑛𝑒𝑔 𝑛𝑒𝑔×𝑛𝑒𝑔=𝑝𝑜𝑠 Same applies to division.

Quickfire Examples 4× −1 =−𝟒 −3 ×7=−𝟐𝟏 −5 × −4 =𝟐𝟎 −8 ÷2=−𝟒 −21 ÷ −7 =𝟑 9÷ −3 =−𝟑 −5 2 =𝟐𝟓 −2 3 =−𝟖 −10−2 2 =𝟏𝟒𝟒 ? ? ? ? ? ? ? ? ?

(Use the front of your diary for blue) Use your diary coloured cards to vote for the correct answer in each question. (Use the front of your diary for blue) RED ORANGE GREEN BLUE

Calculate: −𝟒−−𝟖 4 −4 12 −12

Calculate: −𝟏𝟎 ×𝟓 −5 −50 50 −2

Calculate: −𝟏𝟐÷−𝟐 24 −10 −6 6

Calculate: −𝟑+𝟏𝟑 −𝟑−𝟐 2 −2 10 −10

Calculate: −𝟑× −𝟑 𝟐 −18 −27 27 18

Calculate: −𝟏𝟎𝟎÷ −𝟓−−𝟑 𝟐 −25 25 12.5 50

Exercise 2 Calculate the following: −3× −2 =𝟔 −12÷4=−𝟑 −9 2 =𝟖𝟏 4×−13=−𝟓𝟐 16÷−2=−𝟖 −30 ÷ −6 =𝟓 −8×3=−𝟐𝟒 Determine the missing number: 8×∎=−32 ∎=−𝟒 −6×∎=−18 ∎=𝟑 −8÷∎=−2 ∎=𝟒 −2 2 ×∎=32 ∎=𝟖 −4 2 ÷ −2 2 =𝟒 −3 2 ×−7=−𝟔𝟑 50÷ −5 2 =𝟐 1−−1 × 2−−2 =𝟖 −3 3 ÷ −3 2 =−𝟑 Find two numbers which have the specified sum and product. Sum =−7 Product =12  -3, -4 Sum =1 Product =−12  4, -3 Sum =−3 Product =−10  2, -5 Sum =5 Product =−24  8, -3 Sum =−11 Product =30  -5, -6 Sum =−1 Product =−30  5, -6 Calculate the following: −7+3 −11−−7 =𝟏 −3−−7 2 ×3=𝟒𝟖 −1 100 =𝟏 −2×3 3 − −3−−2 3 =−𝟐𝟏𝟓 −4−−6 (−2−−3) − 3−−7 5+ −10 =𝟒 1 3 a ? b ? c ? a ? ? d b ? e ? c ? f ? d ? g ? ? e ? f 2 a ? 4 b ? ? a c ? d ? b ? 3 c ? ? a ? d b ? ? e c ? d ? e ?