Australian Curriculum Year 5 ACMNA121 Use equivalent number sentences involving mul9plica9on and division to find unknown quan99es Key Ideas •  complete number sentences that involve more than one opera9on by calcula9ng missing numbers, eg 5×□=4×10, 5×□=30−10 •  iden9fy and use inverse opera9ons to assist with the solu9on of number sentences, eg 125÷5=□ becomes □×5=125 •  complete number sentences involving mul9plica9on and division, including those involving simple frac9ons or decimals, eg 7×□=7.7 •  write number sentences to match word problems that require finding a missing number, eg 'I am thinking of a number that when I double it and add 5, the answer is 13. What is the number?' White Fish Thinking •  describe strategies for comple9ng simple number sentences and jus9fy solu9ons •  describe how inverse opera9ons can be used to solve a number sentence •  check solu9ons to number sentences by subs9tu9ng the solu9on into the original ques9on Resources: FISH , inverse poster, 2 die, paper and markers for Spider, Spider Game Vocabulary paXern, increase, decrease, missing number, number sentence (accessing prior knowledge) Introductory Ac?vity Process-­‐Revision Discuss with students how mul9plica9on and division are inverse of each other mul9plica9on and division are part of a fact family if you know your mul9plica9on facts -­‐ you know your division facts. Spider, Spider Game Learning Inten+on: Learners think mathema+cally by using the ‘working backwards’ strategy, rather than repea+ng by rote, mul+plica+on/divison facts. The idea is that they are thinking about the factors that produce the product (or quo+ent). This way they can see the connec+ons between the mul+plica+on facts and division facts. 1.  Divide class into approximately six teams (depending on class numbers) ask learners to suggest the op9mal number. 2.  Teams compete against one another to be the first to draw a complete spider. 3.  Teacher plays the role of “exterminator”. Before the game begins, class and teacher nego9ate what mul9plica9on tables will be prac9sed. This then determines the rules for the game. For example, x3 (body) x6 (8 legs) x8 (two eyes) 4.  Teacher rolls 2 die together to create a 2 digit number. The students then need to work out whether or not that number can be the product that fits with x3, x6, x8. 5.  Eg. Teacher rolls a 4 and a 2. (ie. 42 or 24) -­‐ 4 x 6 = 24 (goes with the 6x tables) Students then can draw the first leg. OR 3x8 = 24 (goes with either x3 or x8) Students can draw body or one eye (not both at same 9me). Game con9nues with teacher repeatedly rolling the dice un9l one team has completed a full spider. Students record the number fact used on the paper. 6.  Students need to be strategic in order to complete their spider. What other number sentences can I make? 7.  When a team is finished -­‐ they call out Spider, Spider. Review aoer play: Does this game help learners understanding of mul9plica9on facts? What might improve how it was played? What could they vary if they played this game again eg. 12 sided dice for division facts. Ac?vity Process The following mini ac9vi9es have been adapted from First Steps and can be used for whole class focus or small group instruc9on. They may also be used as ‘warm-­‐up’ or consolida9on games before other ac9vi9es (depending on the cohort). The idea is that there are mul9ple mini lessons that promote the idea of using mul9plica9on and division to find unknown quan99es Learning Inten9on: Focus on the proper9es of opera9ons and rela9onships between them in order to decide whether number sentences are true. Learners use their understanding of proper9es and rela9onships to complete mathema9cal statements (without finding the ‘answer’ to the calcula9ons), 3 x 4 = 2 x ? Missing Number Have students use inverses to find a missing number. Display 23 x 4 is the same as 92. The inverse 92 ÷ 4 = 23. Hand out a blank strip and ask learners to write a mul9plica9on sentence. Ask: How do you know what the numbers must be? (ini9ally choose larger numbers that discourage students from just calcula9ng the answers.) Invite learners to swap cards and complete the turn around using division 4 × 25 is the same as 100 Equals Have students read the = sign in open number sentences as ‘is the same as’. For example: 32 = 4 x __ and is read as, 32 is the same as 4 mul9plied by ? or 4 lots of. 32 ÷ 4 is 8 equal groups of 4. What does the = sign mean? It does not just mean answer. What does the missing number have to be? Open Sentences Ask learners to create complex open sentences for others to solve. eg. 4 x ? Is the same as ? ÷ 2 or 54 ÷ ? = 6 x ? . Ask learners: How can you make both sides of the = sign equal the same amount? Bigger, smaller or is the same as Have learners generate sets of equivalences. Ask learners: Which changes to the number sentence help calcula9on? Eg. 47 x 19 is ______________47 x 10 400 / 19 is ___________ 400 / 10 47 x 19 is _____________ 47 x 20 400 / 19 is ___________ 400 / 20 Broken Keys Invite students to use connec9ons between opera9ons to deal with a broken calculator key. For example: The division key on my calculator does not work. How could I find the answer to the ques9on 210 ÷ 7? What do you know? Have students use rela9onships between opera9ons to answer: If we know that, what else do we know? Ask them to work out 41 + 41 + 41 = 123. Challenge them to find other number sentences that say the same thing in another way. Eg. 123 / 3 = 41, 123 / 41 = 3, 3 x 41 = 123, 41 x 3 = 123, 123 -­‐ 41 -­‐ 41 -­‐ 41 = 0. Shortcuts Ask the students to check the truth of calcula9ng shortcuts. Eg. Sue said to calculate 57 + 99, she says 56 + 100. To calculate 58 x 99 she says 57 x 100. Is she right? Why or Why not? Explain your thinking using the WHITE FISH – Jus9fying Ques9ons (Defending or valida9ng reasonableness of thinking by providing suppor9ng informa9on) Consolida?on and Extension Ac?vi?es Scootle: L2059 Arrays: Fact Families Make equal rows and columns to explore how numbers can be broken up into factors. For example, the number 24 can be expressed as 12x2 or 2x12, and therefore, it can be divided equally using its factors 12 and 2. Iden9fy a missing factor to complete a factor family. Solve four expressions: two mul9plica9on and two division statements. Recommenda+on: This Learning Object should be used to assist those students who cannot see the inverse rela+onship between mul+plica+on and division. It reverts back to arrays pictures and the 4 fact family number sentences that match the picture. It creates a visual representa+on Stop that Creature! hXp://pbskids.org/cyberchase/math-­‐games/stop-­‐creature/ Hacker has created 10 clones and sent them to destroy the power sta9on! You stop the clones when you name the rule that runs the machine. The game has many levels. Recommenda+on: Teacher to model how the game works first -­‐ can be then used as an independent ac+vity. Technological version of a Func+on (input-­‐output) Machine. Ac?vity Imaginary Learner Give students a set of calcula9ons completed by an “imaginary student” Create a page of ques9ons involving mul9plica9on and division equa9ons. Fill in the answers, ensuring that some are correct and some have errors. Duplicate the page for each learner. When construc9ng the ac9vity, include errors commonly made by the students. Explain that you want them to find the incorrect answers without doing the full calcula9ons. Ask the learners to circle them and jus9fy their responses by annota9ng the page. Background Informa?on Learners should be given opportuni9es to discover and create paXerns and to describe, in their own words, rela9onships contained in those paXerns. This substrand involves algebra without using leXers to represent unknown values. When calcula9ng unknown values, students need to be encouraged to work backwards and to describe the processes using inverse opera9ons, rather than using trial-­‐and-­‐error methods. The inclusion of number sentences that do not have whole-­‐number solu9ons will aid this process. To represent equality of mathema9cal expressions, the terms 'is the same as' and 'is equal to' should be used. Use of the word 'equals' may suggest that the right-­‐hand side of an equa9on contains 'the answer', rather than a value equivalent to that on the leo. Many learners interpret the = sign as ‘makes’ or as a signal to ‘find’ the answer. Asked to complete __ + 7 = 12. they may place a 5 in the blank space but nevertheless say that 12 is the answer. In unpacking this understanding it is important to emphasise that the = sign means ‘is equal to’ and it indicates that both sides of the equa9on represent the same number.