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Year 7 Written Calculations

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Presentation on theme: "Year 7 Written Calculations"β€” Presentation transcript:

1 Year 7 Written Calculations
Dr J Frost Objectives: Be able to add, subtract, multiply and divide numbers of any size (including decimals). Extension: Solve problems involving missing digits within a multiplication/addition. Last modified: 12th April 2016

2 STARTER :: Long Multiplication
Use any preferred method (but preferably long multiplication!) to multiply the following numbers. 284Γ—32=πŸ—πŸŽπŸ–πŸ– 375Γ—65=πŸπŸ’πŸ‘πŸ•πŸ“ =πŸπŸπŸ‘πŸπŸ 739Γ—384=πŸπŸ–πŸ‘πŸ•πŸ•πŸ” 806Γ—608=πŸ’πŸ—πŸŽπŸŽπŸ’πŸ– ? ? ? ? ? This is known as Pascal’s Triangle. Each number is the sum of the two above it. Sadly we will no longer see Pascal’s Triangle for 115 onwards (why?) 11 0 = = = = = If you finish… Put the digits from each calculation into the triangle. What do you notice? 1 1 ? 1 ? 1 ? 2 1 1 ? 3 3 1 1 ? 4 6 4 1

3 3.2Γ—0.04 32Γ—4=128 3.2Γ—0.04=0.128 Multiplying Decimals
First ignore decimal points and multiply as whole numbers. 3.2Γ—0.04=0.128 Correct by counting total number of decimal place jumps.

4 Further Example 84.1Γ—0.05 841Γ—5= Γ—0.05=4.205

5 Test Your Understanding
0.3Γ—0.2 ? πŸ‘Γ—πŸ=πŸ” 𝟎.πŸ‘Γ—πŸŽ.𝟐=𝟎.πŸŽπŸ” 3.21 2 πŸ‘πŸπŸΓ—πŸ‘πŸπŸ=πŸπŸŽπŸ‘πŸŽπŸ’πŸ πŸ‘.πŸπŸΓ—πŸ‘.𝟐𝟏=𝟏𝟎.πŸ‘πŸŽπŸ’πŸ ?

6 11.3βˆ’8.74 11.30 βˆ’8.74 2.56 Decimal Addition/Subtraction
Fill any β€˜gaps’ with 0s. 11.30 βˆ’ Ensure decimal points are lined up so that the digits in each column have the same place value.

7 Check Your Understanding
103.8βˆ’48.91 ? πŸπŸŽπŸ‘.πŸ–πŸŽ βˆ’πŸ’πŸ–.πŸ—πŸ πŸ“πŸ’.πŸ–πŸ— ? πŸπŸ–.πŸ’πŸ‘ +πŸπŸ—πŸ.πŸ•πŸ 𝟐𝟏𝟏.πŸπŸ’

8 Exercise 1 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? (On provided sheet)
Calculate: 3.4Γ—5=πŸπŸ• 4.56Γ—0.2=𝟎.πŸ—πŸπŸ 0.04Γ—0.003=𝟎.𝟎𝟎𝟎𝟏𝟐 48βˆ’13.63=πŸ‘πŸ’.πŸ‘πŸ• 5.48βˆ’1.584=πŸ‘.πŸ–πŸ—πŸ” 42.7Γ—0.56=πŸπŸ‘.πŸ—πŸπŸ 11.3βˆ’7.444=πŸ‘.πŸ–πŸ“πŸ” 45.1Γ—0.0043=𝟎.πŸπŸ—πŸ‘πŸ—πŸ‘ [JMC 2007 Q1] What is the value of Γ—0.4? Solution: 0.42 [Kangaroo Pink 2012 Q1] What is the value of 11.11βˆ’1.111? Solution: 9.999 [JMC 2001 Q8] What is the difference between the largest and smallest of the following numbers? A B C D E Solution: E [IMC 2015 Q1] What is the value of 1βˆ’ βˆ’0.004? Solution: 0.826 [IMC 2002 Q2] Which of the following has the greatest value? A 0.3Γ—7 B 0.5Γ—5 C 0.2Γ—11 D Γ—30 E Γ—100 Solution: D [IMC 2003 Q5] What is the value of ? Solution: [JMC 2012 Q11] In the following expression, each β–‘ is to be replaced with either + or – in such a way that the result of the calculation is β–‘ 45 β–‘ 67 β–‘ 89 The number of + signs used is 𝑝 and the number of – signs used is π‘š. What is the value of π‘βˆ’π‘š? Solution: -1 1 5 ? ? ? ? 6 ? ? ? ? ? ? 7 2 ? ? 8 3 ? 4 ? ?

9 Exercise 1 ? ? ? ? (On provided sheet)
[JMC 1997 Q10] Each day throughout July 1995 I picked 300g of raspberries from my garden. What was the total weight of the raspberries I picked that month? A 10g B 9kg C 9.3kg D 10kg E kg Solution: C [JMC 1998 Q7] The Mystery Prize at the Bank of England Christmas Party was a pile of crisp new Β£5 notes, numbered from up to What was the value of the prize? A Β£116 B Β£117 C Β£580 D Β£585 E Β£1420 Solution: D 11 [JMC 2000 Q10] Each Junior Mathematical Challenge answer sheet weighs 6 grams. If pupils enter the challenge this year, what will be the total weight of all their answer sheets? A 84kg B 840kg C kg D kg E kg Solution: B [JMC 2000 Q14] The DISPUTOR is similar to a calculator, but it behaves a little oddly. When you type in a number, the DISPUTOR doubles the number, then reverses the digits of this result, then adds 2 and displays the final result. I type in a whole number between 10 and 99 inclusive. Which of the following could be the final result displayed? A 39 B 41 C D 43 E 45 Solution: E 9 ? ? 12 10 ? ?

10 Exercise 1 ? ? ? ? (On provided sheet)
[Kangaroo Grey 2014 Q6] Which of the following calculations gives the largest result? A 44Γ— B 55Γ—666 C 77Γ— D 88Γ—333 E 99Γ—222 Solution: B. We can ignore factors of 11 and 111 and just do πŸ’Γ—πŸ•, etc. [JMC 1999 Q21] Granny says β€œI am 84 years old – not counting my Sundays”. How old is she really? A 90 B 91 C D E Solution: D 15 [JMC 1997 Q22] The Grand Old Duke of York, he had ten thousand men, he marched them up to the top of the hill, … By 2pm they were one third of the way up. By 4pm they were three quarters of the way up. When did they set out? A 12 noon B pm C pm D pm E pm Solution: B [TMC Final 2014 Q7] Place exactly three common mathematical operations (which need not all be different) between the digits below so that the result equals You are not allowed to rearrange the order of the digits For example, Γ— 5 βˆ’ 67 Γ— 89. We know this example is wrong because the result is 207 Solution: πŸπŸπŸ‘βˆ’πŸ’πŸ“βˆ’πŸ”πŸ•+πŸ–πŸ— 13 ? ? 16 14 ? ?

11 Exercise 1 ? ? (On provided sheet) 17
[JMO 1999 A7] Before the decimalisation of money in the UK, there were 12 pence (d) in 1 shilling (s) and 20 shillings in 1 pound (Β£). Thus 1 pound 3 shillings and 4 pence was written Β£1 3s 4d. What would have been the total cost of 7 items each costing Β£1 6s 8d? Write your answer in simplest Β£ s d form. Solution: Β£8 [JMC 2003 Q22] Two builders, Bob and Geri, buy bricks at the same price. Bob sells 10 for Β£6 and Geri sells 12 for Β£7. Supposing they sell equal numbers of bricks, what number has each sold when Bob has gained Β£4 more than Geri? A 42 B 60 C 72 D E Solution: E ? 18 ?

12 8Γ·0.2 =80Γ·2 =40 Dividing Decimals ? ? ? ?
It is fine in practice to divide decimals by whole numbers. e.g. Dividing Β£1.68 between 4 people. However, we don’t in general like dividing by decimals. e.g. Dividing Β£3 between 5.6 people. We can’t do that! ? ? ? Consider that for example 8Γ·4=2 and 80Γ·40=2 What does this suggest we can do to both numbers in the division without affecting the result? =80Γ·2 =40 ?

13 1.47Γ·0.3 =14.7Γ·3 04.9 3 | 14.7 2.53Γ·0.011 =2530Γ·11 =230 More Examples
Multiply each number by 10 until we’re dividing by a whole number. =14.7Γ·3 04.9 Ensure decimal point goes in same place in result. 3 | 14.7 2.53Γ·0.011 =2530Γ·11 =230

14 Test Your Understanding
174.9Γ·0.03=πŸ“πŸ–πŸ‘πŸŽ 6.055Γ·0.7=πŸ–.πŸ”πŸ“ ? ?

15 Exercise 2 [JMC 2012 Q2] What is half of 1.01? Solution: 0.505 [Kangaroo Pink 2010 Q1] What is the result of dividing by 2010? Solution: 10,001 [IMC 1999 Q5] 30Γ·0.2 equals Solution: 150 Calculate: a) 22.8Γ·0.5=πŸ’πŸ“.πŸ” b) 5.376Γ·0.06=πŸ–πŸ—.πŸ” c) Γ·0.007=πŸ’πŸŽπŸ‘πŸ”πŸŽπŸŽ d) Γ·1.1=πŸ—.πŸ”πŸ‘ e) Γ·0.11=πŸ—.πŸŽπŸ—πŸ A death laser can fire every seconds. How many times can it fire in 3 minutes? Solution: 45000 6 A rectangle has area cm2 and height 1.3cm. What is its length? 16.34cm What is its perimeter? 35.28cm Calculate: a) Γ·0.16=𝟎.πŸ‘πŸ–πŸπŸ“ b) Γ·1.5=πŸ—πŸ—.πŸ— c) 610.8Γ·0.12=πŸ“πŸŽπŸ—πŸŽ 1 ? 2 ? ? ? 3 7 ? ? ? 4 ? ? ? ? ? ? 5 ?

16 Missing Digit Puzzles 𝐽 𝐽 𝑀 𝑀 + 𝐢 𝐢 𝐽 𝑀 𝐢 ? ? ? ?
This lesson you will be able to work in groups on some puzzles to do with missing digits within column addition/multiplication. [JMC 2007 Q18] The letters 𝐽, 𝑀, 𝐢 represent three different non-zero digits. What is the value of J+𝑀+𝐢 ? A B C D E 15 𝐽 𝐽 Clue: These two digits are the same. Thus J and M must add to 10. Clue: 𝐽 could only be 1 or 2, as greatest result is = 297 ? ? 𝑀 𝑀 + 𝐢 𝐢 𝐽 𝑀 𝐢 Putting this together: 𝑱=𝟏, 𝑴=πŸ—, π‘ͺ=πŸ– Answer is B Clue: We’re adding the same digits in each of the two columns, but get a different digit as the result. There therefore must have been a carry, and hence M is one more than C. ? ?

17 Missing Digit Puzzles Question 1 ?
Get into groups of 3 or 4. Merits to the team with the most correct answers at the end. Effective Team Maths Challenge strategy: Have two people independently work on each problem, and each β€˜initial’ the problem on a shared sheet if you have a solution. When both people have solved the problem, compare your two answers. [JMC 2014 Q12] In this subtraction, 𝑃, 𝑄, 𝑅, 𝑆 and 𝑇 represent single digits. What is the value of 𝑃+𝑄+𝑅+𝑆+𝑇? A B C D E Solution: B ?

18 Missing Digit Puzzles Question 2 ?
[JMC 2003 Q16] In this multiplication each letter stands for a different digit. Which letters stands for 3? A B C D E Solution: D ?

19 Missing Digit Puzzles Question 3 ?
[JMC 2005 Q18] In the subtraction sum on the right π‘Ž, 𝑏 and 𝑐 are digits, and π‘Ž is less than 𝑏. What is the value of 𝑐? A B C 5 D 6 E a number greater than 6 Solution: A ?

20 Missing Digit Puzzles Question 4 ?
[TMC Regional 2011 Q10] 𝐽6𝐾4Γ—7=𝐿9𝑀98 Each of 𝐽, 𝐾, 𝐿 and 𝑀 is a different digit. Find the values of 𝐽, 𝐾, 𝐿 and 𝑀. Solution: 𝑱=πŸ“, 𝑲=𝟏, 𝑳=πŸ‘, 𝑴=𝟐 ?

21 Missing Digit Puzzles Question 5 ?
[TMC Final 2010 Q3] Each of the letters used below is standing for a different single digit. Example: 𝐴𝐡+2=𝐴𝐢 [the answer could be 𝐴=1, 𝐡=3, 𝐢=5 because 13+2=15] If 𝐴𝐡𝐢𝐷×9=𝐷𝐢𝐡𝐴 find 𝐴, 𝐡, 𝐢 and 𝐷. Solution: 𝑨=𝟏, 𝑩=𝟎, π‘ͺ=πŸ–, 𝑫=πŸ— ?

22 Missing Digit Puzzles Question 6 ?
[Junior Kangaroo 2015 Q12] In the sum shown, different shapes represent different digits. What digit does the square represent? A 2 B 4 C 6 D 8 E 9 Solution: C ?

23 Missing Digit Puzzles Question 7 ?
[JMC 2002 Q24] In the multiplication on the right, each letter represents a different digit and only the digits 1, 2, 3, 4, 5 are used. Which of the letters represents 2? A B C D E Solution: E ?

24 Missing Digit Puzzles Question 8 ?
[JMC 2010 Q25] What is the value of 𝑃+𝑄+𝑅 in the multiplication on the right? A B C D E 9 Solution: A ?

25 Missing Digit Puzzles Question 9 ?
[JMC 1999 Q25] The two-digit by two-digit multiplication on the right has lots of gaps, but most of them can be filled in by logic (not by guesswork). Which digit must go in position βˆ— ? A 1 B 3 C 5 D 7 E 9 Solution: D ?

26 Missing Digit Puzzles Question 10 ?
[JMO 2000 B4] How many different solutions are there to the letter sum on the right? Different letters stand for different digits, and no number begins with a zero. J M C + J M O S U M S Solution: 6 ?


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