Patterns and Algebra

Transcription

1 Series Student Patterns and Algebra My name G

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3 Series G Contents Topic Pa erns and func ons (pp. 7) recursive number sequences func on number sequences func on shape pa erns func on machines and func on tables real life func ons func on tables apply the I Do venue solve fabulous Fibonacci and the bunnies solve triangular numbers inves gate Pascal s triangle inves gate Topic Algebraic thinking (pp. 8 5) making connec ons between unknown values present puzzle solve the candy box solve Topic Solving equa ons (pp. 6 ) introducing variables using variables in an equa on simplifying algebraic statements happy birthday solve squelch juiceteria solve Topic Proper es of arithme c (pp. ) Series Authors: Rachel Flenley Nicola Herringer order of opera ons commuta ve rule distribu ve rule equa on pairs apply Date completed / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / Copyright

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5 Patterns and functions recursive number sequences A number pa ern is a sequence or list of numbers that is formed according to a rule. Number pa erns can use any of the four opera ons ( +,,, ) or a combina on of these. There are different types of rules that we can use to con nue a number pa ern: A recursive rule find the next number by doing something to the number before it. A func on rule predict any number by applying the rule to the posi on of the number. Here is an example of a number sequence with a recursive rule. The rule is add 8 to the previous number, star ng with Figure out the missing numbers in each pa ern and write the rule: a b c Rule Rule Rule d 9 8 e 7 5 f 7 Rule Rule Rule What do you no ce about the pa erns a and b in Ques on? Complete these grid pa erns. Look closely at the numbers in the grid and follow the pa ern going ver cally and horizontally: a b c G

6 Patterns and functions recursive number sequences Complete these sequences according to the recursive rule: a Start at and add 7 b Start at 5 and subtract 5 5 c Start at 68 and add Complete these decimal number sequences according to the recursive rule: a Start at.5 and add b Start at 5 and subtract c Start at 0 and add Complete the following number pa erns and write the rule as opera ons in the diamond shapes and describe it underneath. a The rule is b 6 5 The rule is 7 Use a calculator to work out where each pa ern started to go wrong in these single opera on pa erns and circle them. Hint: The first numbers in both are correct. a The rule is b The rule is G

7 Patterns and functions function number sequences There are different types of rules that we can apply to find out more about a sequence: A recursive rule gives the next number by applying a rule to the number before it. A func on rule predicts any number by applying a rule to the posi on of the number. So far we have prac sed the recursive rule to work out the next number in a sequence. Now we will apply the func on rule to this problem: How can we find out the 0th number in this sequence without wri ng out all of the numbers? To use the func on rule we: Use a table like this one below. Write each number of the sequence in posi on. HINT: a good way Work out the rule, which is the rela onship between the posi on of to work out the rule is to see what a number and the number in the pa ern. the sequence is Use the rule to work out the 0th number in the sequence. going up by. This tells you what the first opera on is and Posi on of number 5 0 then you adjust. This sequence is the Rule mes tables moved Number sequence up one so it is +. In each table, find the rule and write it in the middle row. Then apply the rule to posi on 0. a Posi on of number 5 0 Rule Number sequence b Posi on of number 5 0 Rule Number sequence c Posi on of number 5 0 Rule Number sequence HINT: All of these func on rules consist of opera ons: and then + or. G

8 Patterns and functions function number sequences Here is part of a number sequence. Write these numbers in the table provided. This will help you to answer the ques ons below: Posi on of number 5 0 Rule Number sequence Circle true or false for each of the following: a The number in the 6th posi on is true / false b is in this sequence true / false c The number in the 0th posi on is 65 true / false d The number in the 00th posi on is 05 true / false Here is another number sequence but this me of these numbers do not belong. Given the func on rule and the first numbers, use the table below to work out how this sequence should go, then cross out the numbers that do not belong: Posi on of number Rule Number sequence 5 Unscramble the sequence according to this func on rule: 9 6. a Again, use the table below to work out how this sequence should go and cross out numbers that do not belong: Posi on of number Rule Number sequence b What will be the number in posi on 50? G

9 Patterns and functions function shape patterns When you are inves ga ng geometric pa erns, look closely at the posi on of each shape and think about how it is changing each me. How many matchs cks are needed for the first shape? How many more are needed for the next shape? Complete the table for each sequence of matchs ck shapes. Use the func on rule for finding the number of matchs cks needed for each shape including the 50th shape: a Shape number Number of matchs cks Func on rule Number of matchs cks = Shape number + b Shape number Number of matchs cks Func on rule Number of matchs cks = Shape number + c Shape number Number of matchs cks Func on rule Number of matchs cks = Shape number + d Shape number Number of matchs cks Func on rule Number of matchs cks = Shape number + G 5

10 Patterns and functions function shape patterns Gia started to make a sequence out of star and pentagon blocks and recorded her findings in the table as she went. She had to stop when she ran out of pentagons. This is where she got up to: a Help Gia con nue inves ga ng this sequence by using the table below: Shape number Number of stars Number of pentagons 0 Rule for stars Number of stars = Number of pentagons + Rule for pentagons Number of pentagons = Number of stars b How many stars are in the 0th shape? c How many pentagons are there in the 5th shape? Tyson also made a sequence out of pa ern blocks but stopped a er the first shapes and decided to con nue inves ga ng by using the table. Shape number Number of crosses Number of rectangles 0 Rule for crosses Number of crosses = ( + number of rectangles) Rule for rectangles Number of rectangles = ( number of crosses) a How many rectangles will there be in the th shape? b Josie made this shape following Tyson s sequence. What is the posi on of this shape? How do you know? 6 G

11 Patterns and functions function machines and function tables Remember func on machines? Numbers go in, have the rule applied, and come out again. The rule for this func on machine is mul ply by 6. IN 5 9 RULE: 6 OUT Look carefully at the numbers going in these func on machines and the numbers coming out. What rules are they following each me? a b IN 8 RULE: OUT 59 IN 00 RULE: OUT The func on machines showed us that when a number goes in, it comes out changed by the rule or the func on. Func on tables are the same idea the number goes in the rule and the number that comes out is wri en in the table. The rule goes at the top: Rule: + 6 IN OUT Complete these func on tables according to the rule: a Rule: 8 + IN OUT 65 b Rule: 5 IN OUT 6 G 7

12 Patterns and functions real life functions So far we have seen that func ons are rela onships between numbers. These numbers are a ached to real life situa ons everywhere you look. It is possible to create a func on table to show the rela onship between many things, for example: Your high score Live Mathle cs depends on how o en you prac se mental arithme c. The distance that you run depends on how long you run. The amount that you can save depends on how much you earn. The amount of US dollars you get when you travel to Los Angeles depends on the exchange rate. There are many, many more examples. Can you think of any? Complete the func on tables for these real life scenarios: a A pool which fills at a rate of litres every minute. Rule: Number of minutes = Amount of litres Minutes Litres How full is it a er hour? b Maya downloads 5 songs a day onto her MP player. Rule: Number of days = Amount of songs Days Songs How many songs would she have downloaded a er 0 days? c A car is travelling at a speed of 50 km/h. Rule: Number of hours = Amount of km travelled Hours Km travelled How long would it take to travel 800 km? Leah s journey We can show these rela onships on a graph. On the right is a graph of the func on table in ques on c. This is known as a travel graph and shows the rela onship between me and distance. Next, we will look at some examples of graphing func ons. Distance (km) Time (hours) 8 G

13 Patterns and functions real life functions Crawly the caterpillar crawls cen metres per day. This is the graph of my journey shown in the func on table. Plot the points and then join the points with a straight line. Number of days Crawly s journey Distance in cm a Complete the table to show how far he gets in 8 days. b Write a rule for working out the distance if you know the number of days. Rule: Days Distance During the day, Crawly s friend Creepy, crawls 5 cm up a garden wall. At night when he falls asleep, he slides cm back down the wall. a Complete the table below to show how far he gets in 8 days. Number of days Creepy s journey Distance in cm Rule: Days Distance b Write a rule for working out the distance if you know the number of days. Think about the total distance Creepy covers in hours. c Plot the points on the graph above (just like the one in Ques on ), then compare the graphs. How are they different? G 9

14 Patterns and functions real life functions Julie is planning her birthday party and is planning how much food and drink she needs for her guests. She has sent out 5 invita ons. a Complete the table to show how much pizza is needed for different numbers of guests. She has based this table on the es ma on that one guest would eat slices of pizza. b Write a rule in the table for working out the slices of pizzas needed, if you know the number of guests. Rule: Number of guests Slices of pizza c Graph this data by plo ng the points from the table: Pizza catering Do not join the points because the data is about whole slices of pizza not parts of slices. Also you cannot have part of a person, the data is about single people. d How many slices are needed for people? e How did you work this out? Slices of pizza f How could the graph help you? Number of guests g 0 people confirmed they were coming to the party. How many pizzas will Julie need to buy if each pizza has slices? Will there be any le overs? Show your working. 0 G

15 Function tables apply Ge ng ready You and your partner need dice, a pencil and this page. What to do Each player writes their ini als at the top of each column in the scoring tables. For each round, roll the dice for and J. Use the value for and J in the rule. Each player writes the answer in the scoring table which becomes their running score. 5 Players add their scores to the previous score. 6 The winner is the player with the highest score at the end of the round. The overall winner is the player who wins the most points a er rounds. For example If I roll the dice and get for and 6 for J and I am working with ( ) + J, I would calculate ( ) + 6 and my answer would be. So I would write in the first row of the table. The next answer I get I add to and so on un l the end of the table. Round ( ) + J Round ( ) + J Round (6 ) ( J) Total Total Total What to do next Make up your own scoring table where extra points are given for certain answers. You could also decide on a killer number. This number means you wipe out all your points. G

16 The I Do venue solve Ge ng ready A very popular wedding recep on venue has a strict policy in the way they put the tables and chairs together. Below is a bird s eye view of this arrangement. They must only be arranged in this sequence to allow room for their famous ice sculptures in the centre of each table arrangement. What to do Table and Chair Arrangement Table and Chair Arrangement Table and Chair Arrangement Look carefully at the diagram of the floor plan above. a Complete the table below. b Write the rule in the table for the number of tables needed if you know the table and chair arrangement number. c Write the rule in the table for the number of chairs needed if you know the table and chair arrangement number. d Draw what Table and Chair Arrangement would look like in the grid at the bo om of this page. Table and Chair Arrangement Tables 8 6 Chairs Rule for tables Rule for chairs Table and Chair Arrangement G

17 The I Do venue solve What to do next The latest Bridezilla to hire out the I Do venue, wants to know how many guests can fit into the space at this venue. Bridezilla wants to be head of the largest table, which seats 6 guests. This is shown on the floor plan. Work out how many guests she can invite to her wedding by seeing how many will fit in the venue space. The table and chair arrangements must follow the sequence described on the previous page (page ). So, each table arrangement will be a different size. Hint: Try to get 5 more tables in this floor plan. Each table should seat fewer than 6 guests. There should be space between the chairs from all the tables so that guests do not bump against each other when ge ng up from the table. 6 guests Number of guests: G

18 Fabulous Fibonacci and the bunnies solve Ge ng ready A famous mathema cian by the name of Leonardi di Pisa became known as Fibonacci a er the number sequence he discovered. He lived in th century Italy, about 00 years before another very famous Italian, Leonardo da Vinci. His number sequence can be demonstrated by this maths problem about rabbits: How many pairs of rabbits will there be a year from now, if? You begin with one male rabbit and one female rabbit. These rabbits have just been born. A er month, the rabbits are ready to mate. A er another month, a pair of babies is born one male and one female. From now on, a female rabbit will give birth every month. 5 A female rabbit will always give birth to one male rabbit and one female rabbit. 6 Rabbits never die. Month Babies from st Pair Babies from nd Pair Babies from rd Pair Total Pairs of Rabbits Key = pair of rabbits 5 6 What to do Look carefully at the table above to understand the problem. If we kept going, the table would get very wide indeed and quite confusing! So it is up to you to figure out the pa ern. Here is a closer look. Can you see what is happening? What are the next numbers? 5 8 Now, back to the bunnies. Use the table below to answer Mr Fibonacci. How many pairs of rabbits will there be a year from now? Months Pairs of bunnies 5 What to do next Fibonacci now wants to know: How many pairs of rabbits will there be years from now? Use a calculator. Hint: The table below should just con nue from the previous one. Months Pairs of bunnies G

19 Triangular numbers investigate Ge ng ready st nd rd th 5th Write a number sentence for each part of the triangular number pa ern and con nue to complete this list: st = nd = + rd 6 = + + th 0 = th 5 = 6th = 7th 8 = 8th = What to do Let s inves gate a faster way to find the 0th number: Work from the outside in, un l you reach the halfway point adding the numbers. What is the answer each me? Half of 0 is so that means we have 5 lots of, so the 0th triangular number is. What is the 0th number? Work from the outside in, un l you reach the halfway point adding the numbers. What is the answer each me? Half of 0 is so that means we have lots of, so the 0th triangular number is. What to do next Find the 0th triangular number without wri ng down the numbers. Hint ques ons: What are the first and the last numbers? What do they add to? G 5

20 Pascal s triangle investigate Ge ng ready Pascal s triangle is named a er Blaise Pascal and is fascina ng to inves gate because of all its hidden pa erns. Blaise Pascal was born in France in 6 and displayed a remarkable talent for maths at a very young age. His father, a tax collector, was having trouble keeping track of his tax collec ons, so he built his father a mechanical adding machine! (And you thought washing up a er dinner was helpful!) Pascal was actually lucky that this triangle was named a er him as it was known about at least 5 centuries earlier in China. What to do Look carefully at the numbers in the triangle. Can you see how you might go about comple ng it? Once you have worked this out, complete the rest of what you see of Pascal s triangle: HINT: Start with the s at the top of the triangle and add them, you get. 6 Complete the missing sec ons of Pascal s triangle below a b c G

21 Pascal s triangle investigate What to do next Check that the Pascal s triangle on page 6 is correct. Then copy the numbers into the triangle below and colour in all the mul ples of red; hexagons with less than a mul ple of green; and all hexagons with less than a mul ple of blue. 6 Can you see any other pa erns in Pascal s triangle? Look along the diagonals and describe as many pa erns as you can. See if you can find Fibonacci s sequence. G 7

22 Algebraic thinking making connections between unknown values The balance strategy is what we use when we need to find the value of one symbol. Once we know the value of the first symbol, we can find out the value of the second symbol. Clue «+ 0 = 60 Clue «= 00 Use the balance strategy to find the value of ««+ 0 = 60 «+ 0 = 60 0 «= 0 Doing the inverse to the other side of the equa on cancels out a number and makes things easier to solve. This is called the balance strategy. Now we know the value of «we can work out the value of «= 00 0 = 00 0 = 00 0 = 5 Using the balance strategy we do the same to both sides which gives us the answer. Find out the value of both symbols: «a Clue «5 = 5 Clue «= 0 b Clue «9 = 8 Clue «= 96 «««5 = 5 5 = «= = 0 = 0 «9 = 8 «9 = 8 «= «= 96 = 96 = 0 60 = 96 = = 8 G

23 Algebraic thinking making connections between unknown values Now that you have had prac ce following the clues and using the step prompts, try these on your own. Set your work out carefully and always use a pencil so that you can erase mistakes and try again. a Clue «8 = 6 Clue «= 75 b Clue «7 = 9 Clue «+ = 00 Find out the value of both symbols: «It is easier if we put the star on the le hand side. We can swap numbers around with addi on and mul plica on. a Clue 6 «Clue «Steps for finding «+ = 8 = 96 6 «= 8 6 «= Now you can find «6 = «= «= b Clue 9 «= Clue «+ = 00 Steps for finding ««= Now you can find G 9

24 Algebraic thinking making connections between unknown values This me you have clues to work through. There are no step prompts, you are on your own except for one hint: start with the clue where you can find the value of one symbol. Set your working out clearly. Use each box to work out the value of each symbol. Find the value of these symbols. You must look closely at each clue. There are hints along the way. a Clue Clue «= = Clue 6 «= 7 «= = = Clue Clue Clue b Clue 5 «= «= Clue = 60 Clue 5 «= = = Clue Clue Clue 0 G

25 Algebraic thinking making connections between unknown values If you were able to complete the last few pages, then you are ready for the next level of algebraic thinking. This me you have to work a bit harder to find the value of the first unknown. However it is easy if you follow these steps and look very closely at the clues. There are clues within the clues! This page is a worked example. Each step is worked through to help you do this on your own on the next few pages. Find the value of: «Clue tells us that: «+ = + Looking at Clue, we can swap the star and triangle for circles: = 0 So, = 5 Clue tells us that: «= + Looking at Clue, we can swap the star for the triangle plus, so: + + = + Clue «+ + + = 0 Clue «+ = + Clue «= + We know is 5, so: + + = 0 Use the balance strategy + + = 0 So, = + = 6 Now that we know the value of the triangle, we can find out the value of the star with Clue : Clue «= + «= 7 By looking closely at the clues, we have found out the value of all symbols: = 5 = «= 7 G

26 Algebraic thinking making connections between unknown values 5 This page is very similar to the last page. There are step prompts to help you along the way. Find the value of these symbols: = You must look closely at each clue. = There are hints along the way. «= Clue tells us that: «+ = Looking at Clue, we can swap the star and triangle for a circle. Now we have + = = So, = Clue tells us that: «= + 5 Looking at Clue, we can swap the star for the triangle plus 5 so we have: = We know is, so: = Use the balance strategy + = So, = Clue «+ + = 50 Clue «+ = Clue «= = Now that we know the value of the triangle, we can find out the value of the star with Clue : Clue «= + 5 «= G

27 Algebraic thinking making connections between unknown values 6 This me, there are ac vi es where you must use the clues. One has step prompts, the other does not: a Find the value of these symbols: = You must look closely at each clue. «= There are hints along the way. Clue «+ «= Clue «+ «+ = 00 Clue «= 50 Clue tells us that: + = Looking at Clue, this means that: + = So, = We know the value of, so we can put this into Clue : Clue = So, «= b Find the value of these symbols: = You must look closely at each clue. = Clue «+ + = 00 Clue «+ = Clue = + 0 «= Clue Clue Clue G

28 Present puzzle solve Ge ng ready Three students each brought in some presents for the charity drive. Each student spent $6. What to do Can you work out how much was spent on each present? What to do next Label each present with the amount it is worth. G

29 The candy box solve Ge ng ready Miss Harley, the class teacher of 6H, enjoys ge ng her class to think mathema cally by holding guessing compe ons. Her most famous guessing compe on was when she asked 6H to guess the number of cocoa puffs in a bowl if she used 50 ml of full cream milk. In her latest compe ons, she has said that the person who correctly guesses the exact contents of the box gets to take home all the candy. This me she has given clues. What to do Read the clues for Compe on and look carefully at how to solve the problem. This will help you win Compe on. Compe on clues There are 6 candies in the box which are a mixture of choc drops, mallow swirls and caramel dreams. The number of mallow swirls equals four mes the number of choc drops. The number of caramel dreams is equal to the number of mallow swirls. Choc drops Mallow swirls = = Caramel dreams Look at the types of candy as different groups. For every choc drop, there are mallow swirls and caramel dreams. So 9 = 6 That means = So there are: choc drops 6 mallow swirls 6 caramel dreams What to do next Compe on clues See if you could win this box of candy by using these clues to work out the exact contents in the box. Follow the same steps as shown to you in Compe on. There are 8 candies in the box which are a mixture of hokey pokies, pep up chews and chomp s x. The number of pep up chews equals twice the number of chomp s x. The number of hokey pokies equals double the number of pep up chews. Chomp s x Pep up chews Hokey pokies = = G 5

30 Solving equations introducing variables Algebra normally uses le ers of the alphabet to stand for unknown parts of an equa on. These le ers are known as variables and are used in the same manner as symbols such as stars, triangles and boxes. Common le ers used in algebra are: x, y, a, b, c, u and v. = 8 = 8 + = 8 + = 50 Same equa on x = 8 x = 8 + x = 8 + x = 50 Use the balance strategy to find out the value of y: a y + 6 = 68 b y 8 = c y 8 = 7 Using the balance strategy, solve each equa on and then match the le ers to the answers to solve this riddle: What gets we er and we er the more that it dries? The first one has been done for you: O x + 9 = x + 9 = 9 x = 9 x = 5 E y 5 = 9 A a + 7 = 5 W m + 5 = 9 T y + 8 = 5 L 8 + x = G

31 Solving equations using variables in an equation In algebra, variables are used to represent the unknown number or what we are trying to find out. Look at this example: Amity s teacher gave the class a mystery number ques on: The sum of a mystery number and 8 is 6. What is the number? Amity used variable x to stand for the mystery number. She wrote: x + 8 = 6 This is really saying, mystery number plus 8 is 6. Next, Amity used the balance strategy to solve the equa on: x + 8 = 6 x + 8 = 6 8 x = 8 For each ques on, write an equa on using the variable x for the mystery number, then solve it. a The sum of 7 and a mystery number is 6. b A mystery number increased by 5 is 8. c A mystery number doubled is 6. d The difference between a mystery number and 9 is. G 7

32 Solving equations using variables in an equation Find the value of x and y. First find the value of x by using the balance strategy, then you will be able to find the value of y. Show your working out: a x 5 = 5 b x 9 = 7 c x 7 = 8 x y = 50 x y = 8 x + y = 60 x = y = x = y = x = y = Find the length of the side of each of these shapes with algebra. Here you will be using variables to represent the unknown number and the balance strategy to solve the equa on. The first one has been done for you. a If the perimeter of this square is 8 cm, find the length of one side. Call the side x. x = 8 x x = 8 x = 8 x = 7 cm b The perimeter of this pentagon is 0 cm. Find the length of one side. Call the side y. y 5 = 0 y 8 G

33 Solving equations using variables in an equation Algebra can help us find out the value of unknowns or mystery numbers. Look at how this perimeter riddle is solved and then solve the rest in the same way. Call each unknown y. The first one has been done for you. a I am a length between cm and 0 cm. When you add cm to me you get the total length of one side of a square which has a perimeter of 8 cm. What am I? (y + ) = 8 (y + ) = 8 y + y + = 7 y + = 7 y = cm b I am a length between cm and 0 cm. When you add cm to me you get the total length of one side of an octagon which has a perimeter of 0 cm. What am I? y + (y + ) = 0 (y + ) = 0 y + = y = cm c I am a length between cm and 0 cm. When you add 5 cm to me you get the total length of one side of a pentagon which has a perimeter of 0 cm. What am I? y (y + 5) 5 = Read the mystery number riddle, then solve using algebra. Write the informa on as an equa on, use y to stand for the unknown. Show all your working: a I am thinking of a number between and 0. When I add, then divide by and mul ply by 5, I get 0. What is the number? b I am thinking of a number between and 0. When I add 5, then divide by and mul ply by 5, I get 0. What is the number? (y + ) = 0 (y + ) = 0 5 y + = y + = y = G 9

34 Solving equations simplifying algebraic statements An algebraic statement is part of an equa on. Some mes algebraic statements can have the same variable many mes. To simplify a + a + a + a + a, we would rewrite it as 5a. 5a means 5 a which is the same as a + a + a + a + a, but is much easier to work with. Match these algebraic statements by connec ng them with a line: k + k + k + 6 k + 6 k + k + k + k k + k + k + k + k 6k + k + k + 0 8k + 0 Simplify these statements. The first one has been done for you: a k + k + k = 8k b 6x + x + x = c b + b + 5b = d 8y + 5y y = Complete the algebraic addi on stacks. Here is a simple example to start you off. The blocks underneath must add to give the block above. In this example, a + a = 7a. 7a a a a b a b 9a 5b 9b 7b 9a a c d 6x x 5y x 9x y 7y 0 G

35 Solving equations simplifying algebraic statements Remember with algebraic statements, a le er next to a number just means mul ply. 6y means 6 y. You can add and subtract variables that are the same, just like you would for regular numbers 5y + 9y = y 0a 6a = a Use what you know about algebraic statements to solve these equa ons: a a + a = 5 5a = 5 5a 5 = 5 5 a = Use the balance strategy to find out what a is. 5a means 5 a, so with the balance strategy you must do the inverse which is 5 to both sides. b 9b 5b = = = b = c 6c c = 6 = 6 = 6 c = 5 Solve this riddle in the same way as the ques ons above: What is as light as a feather but impossible to hold for long? H 6r r = 8 R 9p p = 5 A 8i + 5i = 9 T 0f f = B 7m + m = 6 E 7x x = 5 Your G

36 Happy birthday solve Ge ng ready Three children are having a birthday party. Can you work out how many candles need to go on each cake? What to do HINT: If Maya and Josh s cakes take up 0 candles what is le for Lim? Read the clues. Show your working. Clue Maya and Josh have 0 candles. Clue Maya and Lim have candles. Clue Lim and Josh have 5 candles. Clue There are candles altogether. What to do next Draw the right amount of candles on each cake: Happy Birthday Maya Happy Birthday Lim Happy Birthday Josh G

37 Squelch juiceteria solve Ge ng ready You work at Squelch Juiceteria, a popular juice bar serving delicious concoc ons to go. The manager has le you in charge of wri ng the daily specials on the board. She has texted you the different juices she wants you to write up but forgot to text the prices and now her phone is turned off. Squelch Juiceteria Specials Mango Tango Strawberry Squeeze Cherry Bliss Apple Berry What to do Use algebra with the clues below to work out the missing prices. Look carefully at this first example and follow the steps to work out the rest. Clue A Mango Tango and a Strawberry Squeeze costs $9. Clue A Mango Tango costs $ more than a Strawberry Squeeze. Use algebra to find out the cost of each. Use m for Mango Tango and s for Strawberry Squeeze. Step Write clues as algebra: Step Combine the clues into one statement to cancel out one unknown: Step Work out the cost of the second juice: m + s = $9 m s = $ m + s + m s = $9 + $ m + s + m s = m + m = $ m = + s = $9 s = Clue A Cherry Bliss and an Apple Berry cost $. Clue A Cherry Bliss costs $ more than an Apple Berry. Use algebra to find out the cost of each. Use c for Cherry Bliss and a for Apple Berry. Step Write clues as algebra: c + a = c a = Step Combine the clues into one statement to cancel out one unknown: Step Work out the cost of the second juice: G

38 Properties of arithmetic order of operations Mr Gain wrote this equa on on the board: =? Max performed the opera on of addi on first, then mul plica on; Amity performed mul plica on first, then addi on. Now they are confused they can t both be right! We need a set of rules so that we can avoid this kind of confusion. This is why for some number sentences we need to remember the rules for the order of opera ons. Rule Solve brackets. Rule Mul plica on and division before addi on and subtrac on. Rule Work from le to right. By following the rules, we can see that Amity was right. Rule says you should always mul ply before you add = = 9 Prac se Rule, doing the brackets first: a 7 + (6 9) = b 8 + ( 7) = c 00 (5 5) = d 0 ( + ) = Prac se Rule, mul plica on and division before addi on and subtrac on: a 00 8 = b = c = d 7 5 = Prac se Rule, working from le to right: a 6 = b = c + 6 = d 7 8 = Check the following sums based on what you know about the order of opera ons. Correct any that are wrong: a 50 (5 9) + 8 = 5 b 00 (7 5) + (0 6) = c (60 8) + (6 ) = 0 G

39 Properties of arithmetic order of operations 5 Make these number sentences true by adding an opera on (+,,, ): Don t forget the order of opera ons! a 96 8 = 0 b 6 = 6 c 8 = d = 5 6 In each word problem there is an equa on frame that solves each problem. Use it to solve the problem: a How much was the total bill if 5 people each had a sandwich worth $8 and people had a drink for $. ( ) + ( ) = b What is the total number of people at a party if invita ons were sent to couples, 7 people could not make it and 5 people turned up unannounced? ( ) + = c 0 children went to the water park. went on the water slides first. The rest went in equal groups to the swimming pool. How many were in one of the groups that went to the pool? ( 0 ) = 7 Work with a partner to see who can get the biggest number in each round. Roll a die mes and write down the numbers in the equa on frame. Compare your answers. The biggest answer wins ten points. The winner is the player with the highest score at the end of Round. Round : ( + ) = Round : + = Round : ( + ) = My Score: / 0 G 5

40 Properties of arithmetic commutative rule Look at The rules say we go from le to right but this sum is easier to answer if we add it like this: (6 + ) +. The commuta ve rule lets us do this when it is all addi on or all mul plica on, no ma er which order we do this the answer will be the same. But this is only if the sum is all addi on or all mul plica on. We can use brackets as a signal of what part of the sum to do first. Look at these examples: = 6 is the same as (7 + ) + = = 70 is the same as ( 5) 7 = 70 Can you go both ways with subtrac on and division? Use brackets to show which pairs you should add first to make it easier: a = b = c = d = Use brackets to show which pairs you should mul ply first to make it easier: a 7 5 = b 6 x 8 = c 50 = d 9 8 = Change the order and use brackets to make these equa ons easier: a = + + = Look for complements. b = + + = Using brackets and changing the order can make it easier to find unknowns. Look at the first ques on as an example, then try the rest. a ( ) 5 = 0 b ( + 6) + = 00 ( 5) = 0 60 = 0 60 = c 0 + (60 + ) = 00 d 8 ( 9) = 6 G

41 Properties of arithmetic commutative rule 5 Let s prac se adding numbers in the order that makes it easier to add. Make a path through each number matrix so that the selected numbers add together to make the total in the shaded box. You can t go diagonally and not all of the numbers need to be used. Start at the bold number: a b c Write equa ons for these word problems. Once you are sure of which opera on to use, order the numbers in way that suits you. a Adele loves reading books. One weekend she read 8 pages on Friday night, 7 pages on Saturday night and pages on Sunday a ernoon. How many pages did she read that weekend? b Two classes competed to see who could raise the most money for charity over days. 6H raised $85 on Monday, $8 on Tuesday and $5 on Wednesday. 6F raised $75 on Monday, $9 on Tuesday and $5 on Wednesday. How much did each class raise? c Luke has been collec ng aluminium cans for a sculpture he is making. He has been collec ng 5 cans a week for the past weeks but s ll needs double this amount. How many cans does he need in total? G 7

42 Properties of arithmetic distributive rule The distribu ve rule says that you can split a mul plica on into two smaller mul plica ons and add them. 5 (50 + ) (50 ) + ( ) 00 + = This comes in handy when the numbers are bigger than normal mes tables ques ons. Fill in the missing numbers for the mul plica ons: a 6 5 = (60 + ) 5 ( 5) + ( 5) + = b 7 5 = (70 + ) 5 ( 5) + ( 5) + = c 56 5 = ( + 6 ) 5 ( 5) + ( 5) + 0 = d 8 6 = ( + ) 6 ( 6) + ( 6) 80 + = Colour match each step of the distribu ve rule. For example, colour the equa on frame labelled in yellow and look for all the parts that match this equa on and colour them yellow too. Then match equa on and so on. By matching all 5 equa ons, you will have the order of the le ers that spell the answer to the ques on below: (0 + 8) 6 (0 ) + (7 ) 0 + = 5 A (0 + 7) (0 6) + (8 6) = 8 N (0 + 9) (0 ) + (9 ) = 57 I (0 + ) (70 ) + ( ) = 6 S 5 (70 + ) (0 ) + ( ) = 76 L What part of a human is in the Guinness Book of Records for reaching the length of 7.5 metres? 5 8 G

43 Properties of arithmetic distributive rule Fill in the missing numbers for the divisions: a 8 = b 08 = You can also use the distribu ve rule with division. (80 + ) ( ) + ( ) + = (00 + 8) ( ) + ( ) + = The distribu ve rule can help us find unknowns if we reverse the first steps. (8 ) + ( ) = 88 (8 + ) = 88 = 88 = 8 Both 8 and are to be mul plied by the diamond so we can rewrite this as shown in line. Then you can use the balance strategy twice to find the value of the diamond. Use the distribu ve rule in reverse to solve this problem: a Over the weekend Blake s dad made 5 batches of cupcakes on Saturday and 7 batches on Sunday. How many were in a batch if the total amount that he made was 80? (5 ) + (7 ) = 80 (5 + 7) = 80 b Jenna and Mel made up a game where if you score a goal you get a certain number of points. Jenna scored 6 goals and Mel scored 5 goals. How many points did they each get if the total number of points was 66? (6 ) + (5 ) = 66 G 9

44 Equation pairs apply Ge ng ready Prac se what you have learned in this topic by playing equa on pairs with a friend. You will need to copy both this page and page, then cut out the cards. copy What to do A er shuffling the cards, place the 8 ques on cards and 8 answer cards face down in separate arrays like this: Player selects one card from each set and if the ques on and answer match, then the player takes both cards and has another turn. If they don't match then Player must return the cards to the same posi on and then it is Player s turn. Con nue un l there are no cards le. The player with the most pairs wins. Both players check through the winner s pairs. 7 (5 + ) G

45 Equation pairs apply (0 + 6) 5 0 (0 7) + (0 ) (5 9) G