p85-130-Gr3-ON-Algebra-Data-Algebra

Unit 2: Algebra (Equality and Inequality) Lesson Content Page Introducing Algebra (Equality and Inequality) 85 87 1 Read Aloud: Safari Park: First Reading 92 96 2 Safari Park: Second Reading 97 100 3 and 4 Investigating Equality 103 104 3 Is It Equal? 108 110 4 Creating Equality 113 115 5 and 6 Investigating Variables 118 121 5 Representing Equal Relationships Using Variables 125 6 Solving for Variables 7 Creating Equality Using Compensation 8 to 10 Properties of Operations 8 Properties of Addition and Subtraction 9 Adding and Subtracting in Parts 10 Properties of Multiplication 11 Using Coding Concepts to Investigate Properties of Operations



Introducing Algebra (Equality and Inequality) PMraotcheesmseast:ical About the tcraPoeonorpmodlrsbemplaseruenomnndviticnsisnaogtgrtl,avi,nritcnegeogf,gln,eisenrceesetleiacncstgiton,innggi,ng In grade three, students investigate equality and determine whether addition, subtraction, multiplication, and division expressions are equivalent. Throughout the process, they also learn more about equivalent relationships for numbers up to 1000, in a variety of real-life contexts. For example, students identify number expressions as equivalent when they produce the same result. They also learn that when a whole is decomposed in several ways, the sum of the various parts are still equivalent. Cathy Fosnot emphasizes the importance of students developing an understanding of part-whole relationships (Fosnot, 2007, p. 5). Working on composing and decomposing is necessary for students to understand equivalence. Caldwell and colleagues suggest that students begin with partitioning a given total so they are focusing on the parts that create the same whole. To help students recognize the patterns related to partitioning numbers, as well as develop an understanding of the part-whole relationship, students should be encouraged to record their combination, look for patterns, and change combinations systematically (Caldwell et al., 2014, p. 19). This helps students understand compensation and equivalence (e.g., 11 + 7 = 10 + 8), and that “taking an amount from one number and ‘giving’ it to the other number (compensating) results in the same sum” (Lawson, 2016, p. 48). Lawson notes that when solving addition problems “initially children are unsure whether reversing the number in an expression will always give the same sum” (Lawson, 2015, p. 48). As students explore and investigate different ways of composing numbers, they also learn about number properties such as equivalence that are evident in the commutative and associative properties for addition and multiplication. Grade three students also investigate variables and how they are used in various contexts. Variables, which are unknown values, are represented by letters or symbols. It is important for students to identify what stays the same (constants) and what changes (variables). Variables can be used in several ways. For example, variables are evident in formulas, when finding the missing addend of a sum, or implicitly evident when working with attributes that can change by their varying characteristics. Algebra (Equality and Inequality) 85

Algebra (Equality and Inequality) Lesson Topic Page 1 Read Aloud: Safari Park: First Reading 87 2 Safari Park: Second Reading 92 3 and 4 Investigating Equality 96 3 Is It Equal? 97 4 Creating Equality 100 5 and 6 Investigating Variables 103 5 Representing Equal Relationships Using Variables 104 6 Solving for Variables 108 7 Creating Equality Using Compensation 110 8 to 10 Properties of Operations 113 8 Properties of Addition and Subtraction 115 9 Adding and Subtracting in Parts 118 Properties of Multiplication 121 10 Using Coding Concepts to Investigate Properties of Operations 125 11 86 Algebra and Data

1Lesson Safari Park: First Reading Language Introduction to the Read Aloud Curriculum Expectations The Read Aloud text introduces math concepts in a meaningful context that allows students to make connections to their everyday lives. During the first Materials: reading of Safari Park, students apply their literacy strategies, such as inferring, using prior knowledge, and synthesizing information, to understand the context Written by Stuart J. of a family trip to Safari Park. (See the Literacy and Mathematics Links chart Murphy in the Overview Guide for more on comprehension strategies.) During the Illustrated by Steve second reading, students become mathematicians and apply the mathematical Bjorkman processes to discover and explore the math concepts embedded in the story. chart paper Both readings are also valuable for assessing where students are, what some of Text Type: Fiction: their misconceptions might be, what concepts need greater emphasis, and what Descriptive—Narrative differentiation may be necessary. Time: 20–30 minutes Oral Communication • 1.4 demonstrate an understanding of the information and ideas in a variety of oral texts by identifying important information or ideas and some supporting details • 1.6 extend understanding of oral texts by connecting the ideas in them to their own knowledge and experience; to other familiar texts, including print and visual texts; and to the world around them Reading • 1.3 identify a variety of reading comprehension strategies and use them appropriately before, during, and after reading to understand texts • 1.5 make inferences about texts using stated and implied ideas from the texts as evidence • 1.7 identify specific elements of texts and explain how they contribute to the meaning of the texts • 2.2 recognize a few organizational patterns in texts of different types, and explain how the patterns help readers understand the texts Assessment Opportunities Observations: Note each student’s ability to: – Use visual cues to make and support predictions – Make inferences and demonstrate understanding by engaging in discussions and follow-up activities – Describe how a variety of text features help them to understand and retell the text Algebra (Equality and Inequality) 87

Read Aloud: Safari Park Summary: The Safari Park amusement park has just opened and Grandpa has 100 tickets to share among his five grandchildren. The rides, games, and treats cost different numbers of tickets, so each of the grandchildren have to plan how they can use their 20 tickets. Suddenly one of the grandchildren, Paul, discovers he has lost his tickets, so the others need to share some of theirs with him. As each child decides how to use the 20 tickets and how to share them with Paul, Paul realizes that he won’t have enough tickets for his favourite ride, the Terrible Tarantula. Disappointed when he receives a ticket for the Rock Toss, Paul carelessly throws it and surprisingly wins 18 tickets. On his next toss, he gets 18 more tickets so everyone, including Grandpa can ride the Terrible Tarantula. NOTE: Choose the prompts that are most suitable for your students. Before Reading Inferring/predicting Activating and Building On Prior Knowledge Building on prior • Show students the front cover and ask what they see. Read the title and the knowledge Connecting names of the author and illustrator. Inferring • Ask them to predict what they think the book might be about and have them explain their reasoning. • Keep track of their answers on chart paper. • Ask if any of the students have been on a safari or to a park with animals. • Ask students what kind of animals they saw and the activities that were available. • A sk whether they needed tickets to do the activities. • Setting a Purpose: Tell the students, “Now that we have made our predictions, let’s read the story together to see what actually happens.” During Reading Analysing • Turn to pages 4–5. Ask students where they think this scene is. (e.g., at a Using text features grocery store/supermarket, market has a “sign”) Predicting Using text features • After reading page 4, ask students what they notice about the words for “Safari Park” and “Terrible Tarantula.” (e.g., They have capital letters.) • Ask students to identify why we use capital letters. (e.g., to begin a sentence, for names, for titles, for something important) • Ask what they learned about Paul on this page. (e.g., He really wants to ride on the Terrible Tarantula.) • Have students predict what will happen next. Record students’ predictions and check back with them later in the story. • Turn to pages 6–7. Before reading the text, have students look at the sketch of Paul and ask them what they notice. (e.g., He looks worried, with sweat around his face.) Ask how Grandpa and the other children are reacting. 88 Algebra and Data

Predicting • Ask students what they think happened (e.g., Paul lost his tickets) and how Inferring Analysing the family will resolve this problem. Predicting • Ask why Abby says “Typical” when Paul realizes he lost his tickets and what Inferring this tells us about Paul. Using text features Analysing/predicting • Have students look at the signs for the different activities and ask them what Using text features they notice about the names. Making connections Inferring • Ask students whether they think Paul will ride the Terrible Tarantula today Analysing/predicting and why they think so. Record students’ predictions and check back later in Using text features the story to see if they are correct. Inferring • Turn to pages 8–9. Ask students why they think Grandpa tells the children Making connections they need to plan which rides they want to go on. Using text features • Ask students how the author and illustrator help us to see the totals for the Using text features different types of activities. (e.g., use of colour, organizes them in a table) • Read the pages. Ask students what ride Chad decides to take Paul on and whether any sibling will take Paul on the Terrible Tarantula. • Read pages 10–11. Ask students what they notice about the word ‘ZOOM!’ on the bottom of page 10 and why it is written that way. (e.g., So, we notice it, and say it louder.) • Ask students whether they have been on a ride like the one Chad and Paul are going on and what it was like. • Read pages 12–13. Note that the author says that Chad and Paul ‘stumbled’ out of their car. Ask students why they think the author chose that word instead of something simpler such as ‘stepped.’ (e.g., ‘Stumbled’ tells us more. They might be dizzy; they might be excited; Maybe the car was hard to get out of; etc.) You could use this example to emphasize that word choice makes stories more interesting. • Ask students what ride Alicia decides to take Paul on. (Elephant Twirl). Ask which child, if any, they think will take Paul on the Terrible Tarantula. • Read pages 14–15. Ask why ‘WHEEEE!’ is written differently. • Ask what students learned from these pages. • Read pages 16–17. Ask students what ride Patrick decides to take Paul on (River Raft) and how they think Paul feels about it. Ask students how they would feel. • Ask students what they notice about the names of all of the rides. (e.g., They each have two words; For River Raft and Terrible Tarantula, both words start with the same letter.) Explain that when consecutive words start with the same letter, it is called alliteration. • Ask students why they think authors use alliteration. Have them think of an example of something in their lives with alliteration. • Read pages 18–19. Ask why the word ‘SPLASH!’ is written differently from the other text on the page. • Ask students who has been on their ride and who has not. Algebra (Equality and Inequality) 89

Predicting • Ask students whether they think Abby will take Paul on the Terrible Making connections Analysing/predicting Tarantula. Record students’ predictions and check back later. Using text features • Read pages 20–23. Ask whether students have ever eaten a pretzel like Abby Analysing/predicting has and how it tasted. Using text features Analysing/predicting • Ask students what game Abby decides to let Paul try. (Rock Toss) Analysing/predicting • Ask what they notice about Paul in the illustrations and what he may be Making connections Analysing/predicting thinking. Using text features • Read pages 24–27. Ask students why some of the words are written in slanted letters (italics). • Discuss what happens when Paul tosses his rock without aiming (he wins 18 tickets) and whether this is probable. • Ask students what they think he will do with the tickets. • Ask how Paul is acting differently on page 26. • Ask students why the word ‘CLANG!’ is written differently on page 27. • Ask students what Paul may do with the tickets now. • Read pages 28–29. Ask students how everyone might be feeling going on the Terrible Tarantula and how students know. • Ask students how they would be feeling if they were going on that ride. • Read pages 30–31. Check back with the predictions that students made throughout the story. • Ask students why the words that were spoken about the ride are written differently. Discuss how they are synonyms for the word ‘said.’ After Reading Synthesizing • Ask students what lesson Paul learned from his experiences. Making connections • Discuss whether students have ever been in a situation that was disappointing. Ask whether it was reasonable for them to be disappointed. Ask how the situation was resolved, if at all. Further Practice Actions and Words... Reader’s Theatre • Reread the story, assigning roles to students. • You may wish to have a pre-discussion about the personalities of each character using cues from the illustrations, dialogue, and reactions to riding the Terrible Tarantula. • Use a narrator and encourage students who are playing the roles of the children to act out the various descriptions you’ve identified together. Writing To develop vocabulary and style in their writing, encourage students to: • Use the idea of ‘alliteration’ (e.g., Terrible Tarantula, Rhino Rides) to create a list of other alliterative titles for rides, games, and activities at Safari Park. 90 Algebra and Data

• Reinforce synonyms for ‘said’ on p. 31 of the book. Use a standard quotation and have students substitute ‘said’ with a variety of synonyms. They read the statements aloud to a partner to see if the partner can identify the replacement word for ‘said’ (see the examples below). – “The park is over here,” said Grandpa. – “The park is over here,” hollered Grandpa. – “The park is over here,” screamed Grandpa. – “The park is over here,” moaned Grandpa. – “The park is over here,” whispered Grandpa. – “The park is over here,” insisted Grandpa. Algebra (Equality and Inequality) 91

2Lesson Safari Park: Second Reading Math Algebra Curriculum Expectations • C2.1 describe how variables are used, and use them in various contexts as Teacher appropriate Look-Fors • C2.2 determine whether given sets of addition, subtraction, multiplication, and division expressions are equivalent or not • C2.3 identify and use equivalent relationships for whole numbers up to 1000, in various contexts Possible Learning Goals • Reflects on the importance of equality in real-life contexts and how it is used to solve problems • Understands that variables represent unknown quantities and constants are quantities that remain the same • Recognizes variables as unknown and explains what the variables represent • Recognizes constants are quantities that stay the same • Represents math expressions using counters • Solves for variables using counters • Understands that both sides of an equation need to be the same or equal • Explains what the equal sign means PMraotcheesmseast:ical About the caPonrnodnbpelercomtvininsgog, l,cvroinemgfl,emrceutiannsigco,antiinngg This story explores the concepts of equality and the use of variables to Math Vocabulary: represent unknown quantities. Throughout the story there are opportunities vubanarkliaannboclweesn,,,eceqoqunuasaltlas,insgtansm, e, for students to represent problems with numbers, variables, and the equal sign, and figure out the unknown in each problem. It is important that students have time to reflect on and explain the math concepts they notice throughout the book, and to communicate their thinking to others. About the Lesson Within the lesson plan, there are more prompts than are feasible to use in one session. Some options for delivering the lesson include: • focusing on some, rather than all the pages, using the pages in between to highlight new concepts in different lessons (e.g., solving equations, looking at unknowns). • carrying out the reading over two or three days, reading a few pages each day, followed by one of the Further Investigation activities. 92 Algebra and Data

Materials: Assessment Opportunities Safari Park, counters, Observations: Throughout the reading, related problem solving, and chart paper discussions, note which concepts are too difficult or too easy for students, so Time: 20–30 minutes next steps can be planned and lessons can be differentiated to meet per session individual needs. Before Reading Reflecting Activating and Building On Prior Math Knowledge • Show the cover and title page of the book and read “Finding Unknowns.” Ask students what they think an unknown is. They can turn and talk to a partner before discussing as a class. • Setting a Purpose: Tell students, “We are going to revisit the story as math detectives and discover interesting things about unknowns and math that we see on the different pages.” During Reading Problem solving • A fter reading pages 4–7, ask students how many tickets they would need to Representing go on all the rides in all the different areas. Problem solving/ reasoning and proving • After reading page 8 (hide page 9), ask students what the unknown is for Reflecting this problem. (e.g., how many Monkey Games Chad can play) Ask what stays Representing the same or is constant. (e.g., the total number of tickets) Reflecting/problem solving/representing • Give pairs of students 20 counters to represent the tickets. Have students Connecting solve the problem using the counters. • D iscuss students’ strategies and have them justify their answers. Ask what operation they used to solve the problem. Write a matching equation and circle the unknown. Explain that before they answer the question, they could put a box or some other symbol to represent the unknown. Explain that the unknown is called a ‘variable.’ • After reading pages 10–11, have students check their representations of counters to see if they match the expressions and whether their solution is correct. • A fter reading pages 12–13 (hide the math on page 12), have students use their counters to represent the rides that Alicia has chosen. • O nce they have completed their representations and found the solutions, have students check and reflect on their answers with the information on the page. • A sk students what the question marks represent. (e.g., the number of rides Alicia can go on in Jungle Kings) Ask what mathematical name they can give to the question marks. Ask what is a constant or what remains the same in the problem. Algebra (Equality and Inequality) 93

Reflecting • After reading page 14–15, have students check to see if they were correct. Reflecting/problem • A fter reading page 16 (hide page 17), have students use their counters to solving/communicating represent the rides Patrick has chosen. When they are done, ask them what Problem solving/ they notice. (e.g., They don’t have enough counters for everything.) Together, representing/ write equations that include the variables. communicating • Read page 17 and have students reflect on their representations. Ask them Representing what they would do to make it only 20 tickets’ worth. Share answers as a Problem solving class. Reflecting/connecting • C over the math on page 18 and read the first sentence. Have students figure Reflecting Problem solving/ out whether Patrick has enough tickets if he doesn’t go on the Tiger Wheel. communicating • Write equations that show the variables. Problem solving/ • A fter reading page 20, have students use their counters to represent the rides communicating/ reasoning and proving and food that Abby has bought. Reflecting • A sk students if Abby has enough tickets left to play any Monkey Games. • Show students page 21 and have them check their representations to see if they match. • Ask students what equation they can write for the problem, including the variable. • A fter reading pages 22–23, students can check their solutions. • After reading pages 24–25, ask whether Paul can go on his favourite ride after winning 18 tickets. Ask how many times he could ride it. • Write any suggested solutions with equations, highlighting the variable. • A fter reading pages 26–27, ask students how many times in total that Paul can go on his ride now that he won 18 more tickets. • Record any suggested equations, highlighting the variables. • Ask students what they think Paul is going to do and why they think so. • Ask students how Paul ended up dividing the tickets. • Discuss whether students think Paul’s decision of what to do with the tickets was fair and why they think so. After Reading Reflecting • Ask students for examples of where they saw math in this story and what Connecting operations were involved. • Ask what variables and constants are. Together, create definitions for the terms. • Ask what the equal sign means. • Discuss where students have seen similar situations in real-life. (e.g., amusement park, a hockey arena, a bowling alley) 94 Algebra and Data

Materials: Building Social-Emotional Learning Skills: Critical and Creative Thinking: Discuss how math is valuable to know in order to solve problems in Digital Slide 28: Safari students’ lives. Ask students what questions they still have about variables and Park Ticket Menu, create an anchor chart. Discuss how mathematicians ask questions about math counters, paper, pencils so they can keep learning. Explain that the class will continue to investigate variables and equality in the weeks ahead. Periodically refer to the chart to see if any of the questions can be answered, or if they have any more questions to add. In this way, students are more aware of the progress they are making. Further Practice • Show Digital Slide 28: Safari Park Ticket Menu. Tell students they have 20 tickets, and they need to decide what rides and/or food they would choose. Have them solve the problem in any way they like and record some of the equations that represent their solutions. Algebra (Equality and Inequality) 95

and3 4Lessons Investigating Equality Math Algebra Curriculum Expectations • C2.1 describe how variables are used, and use them in various contexts as Previous Experience appropriate with Concepts: Students have had • C2.2 determine whether given sets of addition, subtraction, multiplication, previous experiences with working with and division expressions are equivalent or not equality as balance in previous grades. • C2.3 identify and use equivalent relationships for whole numbers up to PMraotcheesmseast:ical 1000, in various contexts Representing, arsacenonofdllmdveiscnpmttgrriouna, vngtceii,oncgsngaieen,tlisepnecrcgott,iibnnrleggea,mtsooonlsing About the Mmaucsanoo,tbnreheasq,tlVaauleonnascctlsseas,dbige,unqvl,auabraryiala,:lbasalnemcse,ed, Algebra is about the study of relationships. The equal sign is one symbol used in mathematics that requires direct instruction so students understand its meaning. Research indicates that, “many students do not recognize that the equal sign denotes equality” (Ontario, 2013, p. 6). Instead, they believe that the equal sign means ‘the answer is.’ It is important that students engage in a variety of activities that represent equality as balance. Marian Small emphasizes that “the equality sign should be viewed as a way to say that the same number has two different names, one on either side of the equal sign” (Small, 2017, p. 374). You may also decide to introduce the not-equal sign (≠) and show how it can be placed between expressions that are not equal. About the Lessons Lesson 3 and Lesson 4 investigate the concept of equality as a balance. Students solve for quantities that make two sides equal and represent the balance using the equal sign. 96 Algebra and Data

3Lesson Is It Equal? Teacher Possible Learning Goals Look-Fors • Recognizes that two sides of an equation are equal when they have the same quantity • Explains that the equal sign means ‘the same as,’ variables are unknown quantities, and constants represent quantities that stay the same • Creates equality by adding or subtracting amounts from either side • Accurately identifies that two sides are equal when they both have the same quantity • Uses addition or subtraction to create equality • Explains what the equal sign means • U nderstands that variables represent unknown quantities • Identifies constants as quantities that stay the same Materials: Minds On (15 minutes) Digital Slides 29–34: • Explain to students that they are going to see some quick images for a short Equal or Not Equal?, Digital Slide 35: Make time. Each image has a Side A and a Side B, and they will determine whether Them Equal, chart paper the two sides are equal or not equal. Time: 60 minutes • Briefly show Digital Slide 29: Equal or Not Equal (e.g., for 5–10 seconds) and then remove it. Ask students what they saw (e.g., both sides have rhombuses and ovals) and whether the two sides were equal. • Show Digital Slide 29 again and discuss whether students’ predictions were correct. Ask students how they can figure out whether the sides are equal without counting the number of shapes. (e.g., Match shapes on both sides, such as the same number of rhombuses.) Discuss how the groupings of the shapes helped to compare the two sides. Ask what remained the same and what changed. • Ask students how many more Side A is than Side B. Ask how they could make the two sides equal. (e.g., Take away six ovals from Side A or add six ovals to Side B.) • Discuss how students could show their strategies using numbers (e.g., 22 − 6 = 16 or 22 = 16 + 6). Ask what the equal sign means. • Write the equation with variables ( + = + ). Ask students what the shapes represent and what the mathematical term is for unknowns. (variables) Together, fill in the variables that show the two sides as balanced. (10 + 6 = 10 + 6) • Repeat this line of questioning for Digital Slide 30: Equal or Not Equal? Create equations with numbers and with variables. Algebra (Equality and Inequality) 97

Working On It (25 minutes) • Show Digital Slide 31: Equal or Not Equal? to the class. Ask students what they see and which side has more shapes. • Show Digital Slide 31 again. Record an expression for each of the two sides, using the different shapes as variables. Together, fill in the numbers in the variables for both sides. Have students turn and talk to a partner about how they could create equality, ensuring there are the same number of each shape on both sides. • Discuss their findings. Below the two expressions, write equations that show different ways to create equality. • Have students work in pairs. Each pair draws their own image similar to Digital Slides 29–34. Each pair then meets with another pair, and they take turns briefly showing their image while the other pair determines whether the two sides are equal. Together, pairs can create an equation that shows how they can make the two sides equal. Differentiation • For students who are having difficulty creating their own set of quick images, have them work with Digital Slides 32–34. Assessment Opportunities Observations: Note whether students can represent the images using variables. You can meet with a small group and create an equation together. Regularly ask what each variable represents. Conversations: Conference with students to ensure they understand equality. Use some of the following prompts: – What does it mean for something to be equal? – How do you know both sides are equal? – Can you show me how both sides are equal? Consolidation (20 minutes) • Meet as a class. Strategically select two or three of the students’ drawings to discuss. Make connections between the variables and what they represent in the drawings. Discuss what changes and what stays the same. • Show Digital Slide 35: Make Them Equal and read through the multiple- choice question together. Review what ‘equal’ means. Clarify that, in this case, it is the number of circles that matters and not the colours of the circles. Have students turn and talk to a partner to solve the problem. Explain that they need to prove that their answer is true and the other choices are not true. • Share answers as a class and have students explain their thinking. • Together, create two expressions, side by side, that represent the different- coloured circles as variables (e.g., different shapes for red and blue to 98 Algebra and Data

represent 16 + 9 and 12 + 12. Ask students how they could adjust the numbers to create equality, so each box has the same number of red circles and blue circles. Record some of their ideas using equations (e.g., 16 – 4 + 9 + 3 = 12 + 12). • Co-create a definition of ‘equality’ and add to the anchor chart previously started with the term ‘variable.’ • Add any new vocabulary to the Math Word Wall with visuals to support the meaning of the words. Further Practice • Independent Problem Solving in Math Journals: Have students draw representations in their Math Journals to show their understandings of the word ‘equal.’ Algebra (Equality and Inequality) 99

4Lesson Creating Equality Previous Experience Possible Learning Goals with Concepts: Students have had • Explains that both sides of an equation are equal when they each have the some experience with representing equality same quantity as a balance using concrete objects. • Writes an equation to represent equality using numbers and variables, and Teacher explains that the equal sign means ‘the same as’ Look-Fors • Creates equality by adding or subtracting in different ways Materials: • Explains that two sides are equal when they both have the same quantity “Creating Equality” • Uses addition or subtraction to create equality and justifies their solution (pages 10–11 in the • W rites equations to show how two sides are equal Algebra and Data big • Accurately uses variables in equations book and little books), • Understands that variables represent unknowns concrete objects, chart • Explains or shows what equality is by giving an example and a non-example paper, markers, counters Day 1 Time: 45 minutes Minds On (15 minutes) 100 Algebra and Data • Show students the “Creating Equality” pages in the Algebra and Data big book and ask them what the pictures show, without referring to the numbers. Ask what a mobile is and whether they have ever seen one before. Explain that the image on page 10 shows a mobile that is unbalanced. Ask how the mobile would look if it was balanced. Ask what would need to be on the other side to create balance. • Draw attention to the legend with shapes and numbers beside the first mobile. Ask students what new information they have. (e.g., The shapes have values; square = 10, circle = 4, triangle = 2.) Ask whether they would need the exact same shapes on each side to balance. (No, because they could represent 2 triangles as 1 circle.) • Have students turn and talk to their partner about how they could balance the two sides. They can use drawings or concrete objects to help them. • Discuss students’ solutions. On chart paper, draw the shapes that students suggest. Create equations for each (e.g., 1 square, 1 circle, and 2 triangles equal 4 circles and 1 triangle: 10 + 4 + 2 + 2 = 4 + 4 + 4 + 4 + 2). Put the shapes around each of the numbers so students can see how the same numbers also have the same-shaped variable. Working On It (15 minutes) • Have students work in triads, with each group getting an Algebra and Data little book. Together, they create equality with the second mobile in as many different ways as they can. They can draw the equivalent mobiles on chart

Materials: paper and fill in the numbers on each shape. They can also create equations to prove that the two sides of the mobiles are equal. “Creating Equality” (pages 10–11 in the Differentiation Algebra and Data big book and little books), • For students who need support, they can represent the numbers within each BLM 32: Make It Equal, chart paper shape with counters so they can see the same number of counters on each Time: 50 minutes side. • For students who need an additional challenge, they can create an additional shape that can be added to the mobile and create a new way of balancing the original mobile. Assessment Opportunities Observations: Pay attention to how students create equality. – Do they add one shape at a time and add up the total each time, or do they have a sense of how many of one shape they will need? – Do they add up one side and make an equivalent total, or do they exchange shapes on one side for equal shapes on the other side (e.g., exchange two triangles for one circle)? Conversations: As students are working, use some of the following prompts to check for understanding: – What is the value of the one side? How will you make it equal? – How do you know when both sides of the mobile are equal? – Is there only one way to make it equal, or can you use different shapes to make it equal? – Are there any shapes that you could exchange for other shapes? How do you know? Consolidation (15 minutes) • Meet as a class. Strategically select solutions that show different ways to balance the mobile. Together, check or create the matching equations for each solution. • Challenge students to look for partial equalities within the mobile (e.g., the two shapes add to 10, so they equal the same as this one shape). • Show students one example and cover two shapes on one side. Ask how they could maintain equality. Discuss how equality can also be achieved by taking away from the heavier side to balance it with the lighter side. Day 2 Minds On (10 minutes) • Show the “Creating Equality” pages from the Algebra and Data big book. • Together, review the ways in which students created equality with the two mobiles on page 10. Make connections between the equations and the visual representations in the mobiles. Algebra (Equality and Inequality) 101

Working On It (20 minutes) • Have students work in partners. Give each pair two copies of BLM 32: Make It Equal. They can balance the mobiles by drawing in the shapes and then creating a number equation for each solution. Encourage them to put the numbers in the corresponding shapes. They can make two different solutions for each mobile. Differentiation • Give fewer balance problems to some students who may need more time focusing on how to create balance. • To offer more challenge, you can have students balance one mobile in as many different ways as possible. • For students who may need a challenge, have them create their own balance mobile, including creating the values for the different shapes. Assessment Opportunities Observations: • Continue with the observations and conversations used on the Day 1 activity. Pay attention to whether students have adapted their strategies from their experiences in the previous lesson. Consolidation (20 minutes) • Have students meet with another pair to compare their solutions for two or three of the mobiles. Each pair needs to prove that they have created equality. Students who created their own mobiles can share them instead. • Meet as a class. Use one of the mobiles to create as many different solutions as possible. Create an equation for each, putting all numbers in shapes that correspond to the ones indicated in the mobile. Highlight how variables of the same shape need to represent the same number. • Building Social-Emotional Learning Skills: Healthy Relationships Skills: Discuss how students worked with their partner to build their mobiles. Ask whether working together made the task easier than solving it on their own. Ask how they worked through a problem if they disagreed. Ask how they can help a partner understand something that may be confusing. Make a list of some of their ideas and refer to the chart when students are having difficulty collaborating. Explain that mathematicians often work with others and share their ideas. By disagreeing and respectfully discussing their differences, they often find better solutions. Further Practice • Independent Problem Solving in Math Journals: Have students create their own mobiles in their Math Journals and explain how the two sides are equal. They can also write matching number equations. 102 Algebra and Data

and5 6Lessons Investigating Variables Math Algebra Curriculum Expectations • C2.1 describe how variables are used, and use them in various contexts as Previous Experience appropriate with Concepts: Students have had • C2.2 determine whether given sets of addition, subtraction, multiplication, previous experience finding the missing and division expressions are equivalent or not number in equations up to 18. • C2.3 identify and use equivalent relationships for whole numbers up to 1000, in various contexts Number • B1.1 read, represent, compose, and decompose whole numbers up to and including 1000, using a variety of tools and strategies, and describe various ways they are used in everyday life PMraotcheesmseast:ical About the Representing, saconolmdvinpmgrou, vnrieincfgale,tcipntrigno,gbr,leeamsoning A variable “is a symbol that can stand for any one of a set of numbers or acnondnsetcratitnegg,iesselecting tools other objects” (Van de Walle and Lovin, Grades 3–5, 2006, p. 307). Van de Walle and Lovin caution that while this definition sounds simple, variables Mvesaqiagrutinaha,bla,Vleedo,xdccpiatoribenousnslst,aaisornyunt:b,, tesryqamuctabiolonl, can be used in different ways, resulting in various interpretations. Students first encounter variables as representing a specific unknown, such as 3 + __ = 10. There is only one answer. Variables can also represent more than one answer, such as 3 + __ > 6. Often, variables are used to represent generalizations, such as the commutative property of addition being a + b = b + a. In the primary grades, students mostly work with variables as unknowns. Shapes are often used as variables rather than letters. Sometimes, two variables are represented within the same equation. Students need to realize that variables of the same shape represent the same value. For example, with the equation ∆ + ∆ + Ο = 10, the two triangles must represent the same amount. In this equation, ∆ can represent 3, and Ο can represent 4 (3 + 3 + 4 = 10). The unknown ∆ can also represent other values; for example, if ∆ represents 4, then Ο must equal 2 for the equation to be true (4 + 4 + 2 = 10). In the given equation, the values of the variables ∆ and Ο change in relation to each other, since the value of one affects the other. About the Lessons Lesson 5 and Lesson 6 give students opportunities to further develop an understanding of variables and missing numbers, using one-digit and two-digit numbers, and to try a variety of strategies. Algebra (Equality and Inequality) 103

5Lesson Representing Equal Relationships Using Variables Teacher Possible Learning Goals Look-Fors • Recognizes symbols (e.g., square, triangle, star, circle) as variables that Materials: BLM 33: Mystery Bag represent missing numbers in equations Riddles, connecting cubes, paper bags, • Solves for the variable using a variety of strategies paper strips Time: 50 minutes • Selects an appropriate strategy to solve for unknowns represented by variables • Explains that a variable represents an unknown number 104 Algebra and Data • Explains or shows the strategy they used • Accurately solves for variables and justifies why their strategy works • Represents variables using symbols Minds On (15 minutes) • Put 23 connecting cubes in a paper bag without students seeing. Show students the bag and explain that it is a mystery bag with a hidden number of cubes in it. Ask them to figure out the number of cubes in the bag by solving Riddle 1 below: – Riddle 1: If I added 10 more cubes to the bag, it would have 33 cubes. How many cubes are in the bag? • Have students turn and talk to a partner. Share answers as a class and have students explain how they got their solution. (e.g., counting back, using facts of ten) • Ask students how they can represent the problem using an equation. Ask how they can represent the missing number. (e.g., use a symbol to represent the variable) Together, create an equation (∆ + 10 = 33). • Repeat this process for the riddles below: – Riddle 2 (25 cubes in a bag): If I subtracted 12 cubes from this bag, it would have 13 cubes. How many cubes are hidden in the bag? – Riddle 3 (35 cubes in a bag): If I added 7 more cubes to this bag, it would have 42 cubes. How many cubes are in the bag? Working On It (20 minutes) • Give each student a riddle card from BLM 33: Mystery Bag Riddles, a paper bag, connecting cubes, and a strip of paper. • Tell students they are going create their own mystery bag using the riddle on their card. Explain that they first must solve their riddle in order to figure

out how many cubes to put in their paper bags. They also will write a matching equation, including a variable, on their strip of paper. • Pair students up and have them read their riddles to their partners. Partners solve each other’s riddles and then check their bags to see if they are correct. They also share the equations they created to see if they agree. Differentiation • Students who need a challenge can create their own riddles instead of using the cards from BLM 33. Assessment Opportunities Observations: Pay attention to how students figure out the hidden numbers. – Are they counting on (e.g., for Riddle 1 in the Minds On, starting at 10 and counting on to 33)? – Are they counting back (e.g., starting at 33 and counting back 10)? – Are they using their knowledge of math facts (e.g., knowing that 10 + 23 = 33 so there must be 23 cubes in the bag)? Conversations: If students are struggling to find the hidden number of cubes, ask some of the following questions: – How many are we adding to/subtracting from the bag? – How can we figure out how many are hidden if we know how many we have in total? Consolidation (15 minutes) • Meet as a class and have students share some of the hidden numbers they found. • Discuss the strategies they used to find their hidden numbers, and create an anchor chart they can refer to as the lessons progress. • Read two or three of the riddles aloud, and have students create the matching equations. Discuss the placement of the variable and what it represents. • Discuss whether the variables represent one specific number, or if there could be more than one possibility. Further Practice • Independent Problem Solving in Math Journals: Have students create a number sentence for their hidden riddle in their Math Journals. Algebra (Equality and Inequality) 105

Materials: Math Talk: “EGGstravaganza” Math Focus: Solving for variables that represent more than one unknown (pages 8–9 in the Algebra and Data big Let’s Talk book and little books), concrete objects, chart Select the prompts that best suit the needs of your students. paper, markers • Show students pages 8–9 in the Algebra and Data big book. What do you Teaching Tip see? What do you think you are supposed to do with the egg carton? (e.g., Fill Integrate the math it with the eggs in the small baskets.) talk moves (see page 7) throughout • Look at the expressions with the symbols. What do you think they mean? Math Talks to maximize student participation What do the shapes represent? What do we call the shapes that represent and active listening. unknown values? (variables) • What do you think the expression is supposed to equal? (18, because there are 18 places for the eggs in the carton.) • What does it mean when two shapes or variables are the same and one is different? (e.g., The numbers that go in the squares are the same and the number that goes in the triangle will be different.) • Look at the eggs in the baskets. How can you fill the egg carton with the eggs in the smaller baskets? You can use more than one of the same basket. For example, there are two eggs in this basket. You can use a group of two eggs twice. You need to find values for the variables with the quantities that will fill the basket. • What did you find? (e.g., 7 eggs, 7 eggs, and 4 eggs) Where would you put these numbers in the expression? Why do the two 7s need to go in the same shape? Put your thumb up if you think this expression is true. How can we make our expression into an equation? (7 + 7 + 4 = 18) • Did anyone find a different solution? (e.g., 8 + 8 + 2) How can we turn this expression into an equation? (8 + 8 + 2 = 18). Are both of our equations true? So, our variables can represent a different set of numbers. Partner Investigation • Let’s look at the next expression. How is it different? (e.g., This time, you need three different numbers to fill the egg carton.) How is the third expression different? (e.g., This time you are multiplying two different numbers together to equal 18.) • Work with your partner to find several solutions for the two expressions. You can use the little books and concrete objects or drawings to help you. Record your equations on chart paper. Be sure to put your numbers inside the shapes that represent variables. Follow-Up Talk • What did you find? Share some of the students’ solutions and have them prove that they are correct. 106 Algebra and Data

• What have we learned about variables from this Math Talk? (e.g., Shapes or symbols that are the same must have the same number; a shape can represent more than one unknown quantity.) Further Practice • Repeat the Math Talk, focusing on the egg carton that holds 24 eggs. Algebra (Equality and Inequality) 107

6Lesson Solving for Variables Teacher Possible Learning Goal Look-Fors • Solves for variables in addition and subtraction equations, using a variety of Materials: Digital Slide 36: strategies Make It True, BLM 34: Variable Cards, various • Explains or shows that the equal sign means ‘the same as’ or ‘balanced’ concrete objects • Explains that the variable represents a missing number Time: 50 minutes • Selects an appropriate strategy to solve for a variable • Explains or shows the strategy they used to solve for the variable 108 Algebra and Data • Understands that addition and subtraction are inverse operations Minds On (10 minutes) • Show students the multiple-choice question on Digital Slide 36: Make It True. Students can turn and talk to a partner about which answer is correct, and why the other choices are not correct. • Share answers as a class and have students explain their thinking. Review what the box symbol is (a variable) and what it represents (an unknown number). Ask whether the variable can represent more than one answer. • Discuss students’ strategies, highlighting whether they used addition or subtraction in their calculations. Working On It (20 minutes) • Students work in pairs. Give each pair a set of cards from BLM 34: Variable Cards. • Students solve for the variable in each problem, using concrete objects, drawings, or mental math strategies. • When they are finished, have students develop their own equation that has a variable. Differentiation • Vary the number of problems to meet the needs of your students. • You can use the blanks on BLM 34 to create your own variable cards using numbers with sums up to 1000. Assessment Opportunities Observations: Pay attention to how students are solving for the missing number. – Are they counting up or back from the given values (e.g., for 2 + 5 = x + 3, counting on from 3 to get to 7)? – Are they using guess and check, or do they have a systematic strategy?

– Are they using compensation and equivalence (e.g., for 2 + 5 = x + 3, students recognize that 3 is 1 more than 2, so they would need to take 1 away from 5 to find the missing number)? – Are they using the commutative property (e.g., for 6 + 5 = x + 6)? Conversations: If students are solely using guess and check, pose some of the following prompts: – Look at each side of the equation. Are any of the numbers close in value? If this number is 1 more than the number on the other side, how might that affect the unknown? – What do the numbers on this side equal? What does the other side need to equal? Do you need to add a small quantity or a large quantity to the number on the other side of the equation? Is there a math fact that can help you estimate what the unknown might be? Consolidation (20 minutes) • Students meet with another pair to check their work. They can exchange the equations they created and solve each other’s problems. • Meet as a class. Use one of the problems to highlight the various strategies students used. • Discuss how and why students used different strategies for different problems. Further Practice • Independent Problem Solving in Math Journals: Have students choose two of the following missing number sentences to represent and solve in their Math Journals. 24 + 15 = 40 − □ ∆−5=7+3 □ + 6 = 18 − 5 58 − 7 = ∆ + 3 Algebra (Equality and Inequality) 109

7Lesson Creating Equality Using Compensation Math Algebra Curriculum Expectations • C2.1 describe how variables are used, and use them in various contexts as Teacher appropriate Look-Fors • C2.2 determine whether given sets of addition, subtraction, multiplication, and division expressions are equivalent or not • C2.3 identify and use equivalent relationships for whole numbers up to 1000, in various contexts Possible Learning Goals • Creates equality using a variety of strategies, including compensation • Writes equations to represent the equality, using variables to indicate the unknown values • U nderstands what equality is and what variables represent • Creates balance using more than one strategy • Writes a number equation that matches the strategy used to create equality • R epresents the unknown quantity using a variable • Explains how their strategy creates equality Minds On (20 minutes) • Use the following Math Talk as the Minds On activity. Materials: Math Talk: “Creating Equality” Math Focus: Investigating creating equality using compensation (pages 10–11 in the Algebra and Data big Let’s Talk book and little books), chart paper, markers Select the prompts that best meet the needs of your students. Time: 60 minutes • Show the spread “Creating Equality” in the Algebra and Data big book and 110 Algebra and Data draw attention to the first pan balance at the top of page 11. What do you see? (e.g., bags with the word ‘Marbles’ on them; a pan balance) • What do we know about the two sides of a pan balance when they are level? (e.g., Both sides have the same mass.) What does the pan balance look like if it is not balanced? What do you know about the two sides and which one is heavier? Are these two sides level? What do you know from the position of the pan balance?

• What do you think you need to do? (e.g., find the number of marbles for Bag D on the right side that will create equality) Turn and talk to a partner. • What did you find? (e.g., We added 17 + 22, which is the number of marbles on the left side and subtracted the 12 marbles on the right side, so 39 – 12 = 27.) How many marbles are on each side? (39) What equation can we make to show what you did? (17 + 22 = 12 + 27). Write the equation on the board. Which is the variable? (27) Let’s put a box around it. • Did anyone use a different strategy? (e.g., We saw that there were 10 more marbles in Bag B on the left than in Bag C on the right, so Bag D has to have 10 more marbles than Bag A to balance it out. So Bag D has 27 marbles.) Did you need to figure out the total number of marbles on both sides? Why? (No, because we adjusted the partial amounts on each side.) What operation did you use to create balance? (addition) What equation can we write? (17 + 22 = 12 + 27) So, your equation is like the first equation. • Did anyone have a different solution for the variable? (e.g., There are 10 more marbles in Bag B than in Bag C, so we split those in half and put 5 in Bag C. Now they both have 17. Bag D needs to have the same as Bag A so it has to have 17.) Did you need to know the total number of marbles on each side? What equation represents your thinking? (17 + 22 – 5 = 12 + 5 + 17) You used compensation. You took away marbles on the left side and put them on the right side. What operations represent the actions? (Subtract from one side and add the same amount on the other side.) Could you make Bag A and Bag D equal by moving marbles from one bag to the other? (No, because you can’t equally divide 17, which is an odd number.) So, you used compensation for one of the bags and then added more marbles. • Did anyone have another solution? (e.g., We took 5 marbles out of Bag A and put them in Bag D, so now Bag A and Bag C both have 12 marbles. Now Bag B and Bag D need to have the same amount. Bag B has 22 marbles and we already put 5 in, so we need to put in 17 more marbles.) How can you represent what you did with an equation? (17 – 5 + 22 = 12 + 5 + 17) You used compensation too by subtracting from one side and putting the same amount on the other side. • Do you think that you can use compensation in other problems? Let’s investigate. Working On It (20 minutes) • Show students the three remaining pan balances on page 11 in the big book. Have them create equality for the pan balances in a variety of ways. Encourage them to use compensation for at least one of the solutions. • Students can record their findings on chart paper and write matching equations for their solutions. • In pairs, have students find the missing number using compensation. They can write equations to match their actions. Algebra (Equality and Inequality) 111

Differentiation • Adjust the number of problems students solve according to the needs of your students. Assessment Opportunities Observations: Pay attention to whether students can use compensation or whether they just add marbles to the unknown side. Conversations: If students are having difficulty solving the problems using compensation, pose some of the following prompts: – How could you create equality if you can’t add any more marbles? Imagine that you have 8 cookies and your friend had 12 cookies. There are no more cookies. How could you make sure that both of you have an equal number of cookies? Consolidation (20 minutes) • Meet as a class. Together, discuss the various problems. Focus in on the solutions that involved compensation. • Ask students whether compensation could be used with all the problems. (e.g., Yes, we used compensation for at least one of the bags in each problem.) • Ask if anyone was able to balance without adding any more marbles. (e.g., No, because the total number of marbles in all cases were odd). • Co-create equations for some of the students’ solutions. Ask where the variables are. Highlight how they use the same variable to represent taking one quantity from one bag and adding the same amount to another bag. Ask why it is important to have a different variable for each different number. • Conclude with a discussion about how compensation is a useful strategy for creating balance as it helps to find the missing number in some number sentences. Emphasize that compensation is just one of many strategies that they can use when problem solving, and their choice usually depends on the type of problem they are solving. 112 Algebra and Data

to8 10Lessons Properties of Operations Math Algebra Curriculum Expectations • C2.1 describe how variables are used, and use them in various contexts as Previous Experience appropriate with Concepts: Students worked with • C2.2 determine whether given sets of addition, subtraction, multiplication, mental math strategies and equations with and division expressions are equivalent or not numbers up to 50 in grade two. • C2.3 identify and use equivalent relationships for whole numbers up to 1000, in various contexts Number • B2.1 use the properties of operations, and the relationships between multiplication and division, to solve problems and check calculations • B2.3 use mental math strategies, including estimation, to add and subtract whole numbers that add up to no more than 1000, and explain the strategies used PMraotcheesmseast:ical About the Representing, saconolmdvinpmgrou, vnrieincfgale,tcipntrigno,gbr,leeamsoning In grade three, students investigate the properties of the operations and the acnondnsetcratitnegg,iesselecting tools relationships between multiplication and division. They can also apply this understanding to solve problems and develop effective mental strategies MmsesrvuNaqeniota‘gbbhaulteOrranetheesaicrTta,s,aelisEtVi,boolcnesheo:slnctsvreasaicSmisaeosarsmhaterrnt,usisibidlp,eavdeyu‘s,eidie,anlnfue,adna’vsebsqritsbe,tteytsuisuurero:dasaoddtqnetcctoeuti,o’itohnaa,anonetultdrosi,ssvdteeae, ,rcean for carrying out calculations. As Marian Small explains “the idea is not to ready. memorize the principles by name, but to be so familiar with our number system that they are used naturally and informally” (Small, 2009b, p. 108). Some of the properties are described below. • Inverse Relationships: Addition and subtraction are inversely related, as are multiplication and division. In other words, they are opposite to or ‘undo’ each other. • Commutative Property: With addition and multiplication problems, the numbers can be operated in any order (4 + 5 = 5 + 4; 4 × 5 = 5 × 4). This is not true for subtraction or division. • Associative Property: With addition and multiplication, the numbers can be regrouped and added or multiplied without changing the outcome of the operations (4 + 5) + 6 = 4 + (5 + 6); (4 × 5) × 6 = 4 × (5 × 6). • Decomposing in Addition and Subtraction: You can add and subtract in parts. For example, you can add 26 + 14 and 26 + 10 + 4, or you can subtract 47 − 23 as 47 – 20 − 3. continued on next page Algebra (Equality and Inequality) 113

• 0 and 1 Principles: When 0 is added or subtracted from a number, the answer is the original number (identity principle). When multiplying a number by 0, the answer is 0. When 1 is added to or subtracted from a number, the answer is the next or the previous number in the counting sequence. When you multiply or divide by 1, the answer is the original number (identity principle). About the Lessons In the following lessons, students investigate some of the properties of addition, subtraction, multiplication, and division. In some cases, they create conjectures based on limited examples and then test them out to see if they are rules that can be applied to all numbers. Lesson 10 is comprised of a series of Math Talks that address the properties of multiplication. You may decide to revisit these Math Talks when students are studying multiplication concepts in the Number strand. This is a good opportunity to model and reinforce the Mathematical Modelling Process and its four components, which include: • Understand the Problem • Analyse the Situation • Create a Model • Analyse and Assess the Model Use an anchor chart of the four components to highlight how your students move back and forth and between them as they develop, refine, and test their models. For example, as students test their models, they may need to revisit the problem for further clarification (Understand the Problem), or reanalyse the conditions related to the problem (Analyse the Situation) in order to select more effective strategies and tools. Lesson 9 has some suggestions on how and when to reinforce these ideas; they will need to be adjusted to reflect how your students are progressing through the process. 114 Algebra and Data

8Lesson Properties of Addition and Subtraction Teacher Possible Learning Goals Look-Fors • Writes equations to represent concrete models of equal sets • Explains why the commutative property of addition is true • Understands and explains the inverse relationship between addition and subtraction • Describes the inverse relationship between addition and subtraction • Accurately combines or partitions the elements (terms) of an equation in different ways, using addition and subtraction (e.g., 13, 5, and 18 can be represented as 13 + 5 = 18, 5 + 13 = 18, 18 − 13 = 5, 18 − 5 = 13) • Represents the inverse relationship of addition and subtraction using concrete objects and models Materials: Minds On (20 minutes) Digital Slide 37: Write • Project Digital Slide 37: Write an Equation. Ask students what they see and an Equation, different- coloured connecting what they wonder about. They can turn and talk to a partner. cubes, BLM 15: Number Lines, paper strips • Have students share with the group. (e.g., There are four green cubes and five Time: 60 minutes red cubes. The top and bottom parts of the picture have the same number of cubes [nine]. The bottom cubes are just separated.) • Ask what equations students could use to represent the picture. • Record students’ equations on an anchor chart (e.g., Addition: 4 + 5 = 9, 5 + 4 = 9, 9 = 4 + 5, 9 = 5 + 4; Subtraction: 9 − 5 = 4, 9 − 4 = 5; students might also show 4 = 9 − 5, 5 = 9 − 4). • If students use only addition equations, ask if there is another operation that is evident in the picture. • Ask students what they notice about all of the equations. (e.g., They use the same numbers, but they are in a different order or use the opposite operation.) • Discuss how addition and subtraction are related. (e.g., They are opposite; they undo each other.) Have one student use connecting cubes to demonstrate the action of building 4 + 5 and have another student demonstrate the action represented in 9 − 4. Ask how the actions of putting together and breaking apart represent addition and subtraction. • Ask how students can use real connecting cubes to prove that 4 + 5 = 5 + 4. Turn the connecting cubes around so students can see how 4 + 5 can become Algebra (Equality and Inequality) 115

5 + 4 just by rotating it and not adding or subtracting any cubes. Explain that 4 + 5 = 5 + 4 is an example of a ‘turn around fact.’ • Build a train of connecting cubes with 11 red cubes and 4 green cubes. Ask students how they can represent the cubes using addition and subtraction equations. Have students turn and talk to a partner about what they see and how they could represent it using addition and subtraction equations. • Together, discuss their findings. Discuss how addition and subtraction are opposite operations. Working On It (20 minutes) • Students work in pairs. Provide each pair with connecting cubes and number lines from 0–20 or from 0–50 from BLM 15: Number Lines. • Students create a concrete model of related addition and subtraction facts and then also represent them on a number line. They then create the four equations that represent their models on separate strips of paper. Differentiation • For students who may have difficulty creating their own models, provide either a concrete model or a number-line model from which to create the four equations. Assessment Opportunities Observations: Pay attention to how students are organizing their equations. – Are they able to represent the operations with the correct signs? – Can they write the inverse operations from the first equation, or do they need to model them concretely? – Do they understand that the total can be on the left-hand side of the equal sign (e.g., 9 = 4 + 5)? Conversations: Pose some of the following prompts to further probe students’ thinking: – How did you create that equation? How does it match your model? – Why did you use subtraction for this equation? Model what action it represents using your concrete model. – Are both sides of the equal sign the same? How do you know? – How did you reorganize the parts of the equation? Consolidation (20 minutes) • Have students meet with another pair. Pairs take turns showing their models and having the other pair think what the matching equations are. They can then verify each other’s responses. 116 Algebra and Data

• Discuss the inverse relationship of addition and subtraction. Review the commutative property and have students prove that 4 + 6 = 6 + 4. • Create definitions for the inverse relationship and the commutative property of addition and subtraction. Students do not need to know the names of the properties, but they need to understand how and why they are true. • Building Social-Emotional Learning Skills: Positive Motivation and Perseverance: Assure students that they will practise writing equations in the upcoming lessons. Discuss how practice is essential in order to fully understand math concepts. Explain how it is like practising a sport for an upcoming game. Just like your body gets stronger from working out, your brain grows with practice. Further Practice • Independent Problem Solving in Math Journals: Have students choose a two-digit number. In their Math Journals, they generate as many addition and subtraction equations (inverse) as they can in which their chosen two- digit number is the ‘total.’ If they choose 20, for example, they might come up with the following equations: 10 + 10 = 20; 20 − 10 = 10; 20 = 15 + 5; 20 = 5 + 15; 20 − 5 = 15; 20 − 15 = 5; etc. Algebra (Equality and Inequality) 117

9Lesson Adding and in Parts Subtracting Teacher Possible Learning Goal Look-Fors • Decomposes and recomposes numbers when adding and subtracting, using Materials: chart paper, base ten mental math strategies blocks (optional) Time: 50 minutes • Decomposes numbers into more friendly numbers • Recomposes numbers (e.g., 24 + 46 is like (24 + 6) + (46 − 6) or 30 + 40) • Explains why the answer is the same when you add or subtract using different parts • Explains mental strategies for adding and subtracting numbers Minds On (15 minutes) • Print “36 + 27” on chart paper in a horizontal format. Have students turn and talk to a partner and discuss mental math strategies for solving the problem. • Have students share their responses. Record each answer on the board. Possible strategies include: – Add the tens as 30 + 20; add 6 + 7; then, add 50 + 13. – Keep 36 whole; add 20, which equals 56; add 4, which equals 60; add 3. – Add 4 from the 27 to the 36, which equals 40; add the 20 from the 23, which equals 60; add the remaining 3. • Ask students whether adding in parts affects the total and why they think this might be. (e.g., No, because you are still adding or subtracting the same amount, but just in bits at a time.) Comment that this is an interesting conjecture or idea that can be investigated further to see whether it can be generalized into a rule. Explain that they will further explore adding in parts with larger numbers. (Understand the Problem) • Co-create the conjecture that numbers can be added in parts without affecting the sum. Ask what students need to do in order to determine whether the conjecture is a rule. (e.g., Try it with other numbers.) (Analyse the Situation) • Ask students whether they think they can subtract numbers in parts. Tell students that they can explore this conjecture, too. Write a conjecture for subtracting in parts. (Understand the Problem) • Ask students what they need to do to prove their conjecture. (e.g., Test it and prove that it works with other numbers.) (Analyse the Situation) 118 Algebra and Data

Working On It (Whole Group) (20 minutes) • Tell students that they need a model to test out their conjecture about adding in parts. Ask what numbers they could use and how they could test them out. Let students select the numbers and strategies. (Create a Model) • As an example, students may suggest adding with a three-digit expression (e.g., 324 + 436). Possible strategies include: – Add the hundreds, then add the tens, and then add the ones. – Add on from 324, first with the hundreds, then the tens, and then the ones. – Add on from 436 since it is the larger number. – Add 6 from the 436 to 324, which equals 330, then add 330 + 430. • Do two or three more examples suggested by students. After each test, ask students whether they think the conjecture is true. • Investigate subtracting in parts by having students suggest a mental math problem. As an example, students may suggest 652 − 237. (Create a Model) Students can turn and talk to a partner about possible strategies. Possible solutions include: – From 652, subtract 2 hundreds, which equals 452; subtract 3 tens, which equals 422; subtract 2 ones to equal 420; and then subtract 5 ones to give 415. – Subtract 2 from each number so it is 650 – 235; subtract 200; and then subtract 35. – Add up numbers from 237 to get 652, first adding 400 to 237 to equal 637; add 3 to 637 to equal 640, then add 12 to 640 to equal 652. Total the added parts: 400 + 3 + 12 = 415. • Repeat with two or three more examples suggested by students. Discuss whether students think you can always subtract in parts. Differentiation • Adjust the numbers used in problems to meet the needs of your students. • For students who struggle with decomposing numbers, meet with them in a small group and use base ten blocks to practise decomposing and recomposing numbers in various ways. Assessment Opportunities Observations: Pay attention to whether students are able to mentally decompose numbers. Consolidation (15 minutes) • Revisit the two conjectures. • Ask students whether they think the conjecture about adding in parts is true. Explain that it seems to be true, but real mathematicians may test it in other Algebra (Equality and Inequality) 119

Materials: ways before making a conclusion. Ask what they could do differently or in number cubes addition to their test. (Analyse and Assess the Model) • Discuss whether subtracting in parts affects the answer. Explain that their conjecture seems to be true for the numbers that they have worked with so far. • Discuss why it makes sense that you can add and subtract in parts. (e.g., You are adding or subtracting the same amount, but doing so in smaller chunks.) Further Practice • Independent Problem Solving in Math Journals: Have students roll two number cubes twice to create a new addition sentence in their Math Journals. Encourage them to complete the addition using strategies from today’s lesson. 120 Algebra and Data

10Lesson Properties of Multiplication Teacher Possible Learning Goals Look-Fors • Uses the array to investigate repeated addition, multiplication, division, and the commutative property of multiplication • Explains why the commutative property of multiplication is true • Understands and explains the inverse relationship between multiplication and division • Demonstrates an understanding of the zero and identity properties in multiplication • Connects previous knowledge of equal groups and repeated addition to multiplication • Understands, explains, and uses the commutative property of multiplication • D escribes the inverse relationship between multiplication and division • Represents the inverse relationship of multiplication and division using concrete objects and models • Accurately records the equations that represent the relationships • Uses concrete objects or a diagram to represent the zero and identity properties About the Lesson This lesson is composed of Math Talks that address the inverse relationship of multiplication and division, as well as some properties of multiplication. You may want to revisit these talks when students are learning multiplication concepts in the Number strand. Teaching Tip Math Talk 1: Integrate the math Math Focus: The inverse relationship of multiplication and division, and the talk moves (see commutative property of multiplication page 7) throughout Math Talks to maximize Let’s Talk student participation and active listening. Select the prompts that best meet the needs of your students. continued on next page Algebra (Equality and Inequality) 121

• Draw a 4 × 6 array of circles on the board. What do you see? What do we call a group of objects arranged in this way? (an array) Where are the columns and where are the rows? • Turn and talk to your partner about how many circles you see and how you know. • What did you find? Did you need to count each circle? Why? What strategy did you use? (e.g., repeated addition: 4 + 4 + 4 + 4 + 4 + 4 = 24; 6 + 6 + 6 + 6 = 24) • How can you represent this array using multiplication? (e.g., 4 × 6 = 24) What does that mean? (e.g., There are 4 groups of 6.) Where are the 4 groups? Who saw it differently? (e.g., 6 × 4 = 24) What does that mean? (There are 6 groups of 4.) Where are the 6 groups? Does the order of the numbers affect the answer when we multiply? • We learned about the commutative property for addition. Do you think it works for multiplication, too? How can we use the array to prove that 4 × 6 = 6 × 4? (e.g., Turn the array around so the rows are the columns and the columns are the rows.) • We can call 4 × 6 and 6 × 4 turn around facts. The answer is the same, but do they represent the same thing? Let’s think about 4 packages of 6 pencils and 6 packages of 4 pencils. What is the same and what is different? (e.g., Both have 24 pencils, but the groups are different.) • What other operations can be represented in the array? (e.g., division) • Where do you see division? (e.g., I see 24 circles divided into 4 groups, which equals 6 circles in each group.) How can we write this as an equation? (24 ÷ 4 = 6) Did anyone see it differently? (24 ÷ 6 = 4) • Let’s look at the equations we created for multiplication and division. How are they the same and how are they different? (e.g., All contain the same numbers, but the operations are different.) How are the operations of multiplication and division related? (e.g., They are opposite operations; one undoes the other.) What other operations do we know that ‘undo’ each other? (addition and subtraction) Teaching Tip Math Talk 2: Integrate the math Math Focus: The associative property of multiplication talk moves (see page 7) throughout Let’s Talk Math Talks to maximize student participation Select the prompts that best meet the needs of your students. and active listening. • Show students the multiplication expression 2 × 3. What operation is used in this expression and what does it mean? (e.g., It’s multiplication and it means 2 groups of 3). How can I draw 2 × 3 using an array? Draw the array. 122 Algebra and Data

• Look at this multiplication expression 2 × 3 × 4. According to this expression, how many of the arrays do we have now? (four) Draw four 2 × 3 arrays beside each other in a row. What is the product and how do you know? (e.g., 6 + 6 + 6 + 6 = 24; double 6 and double it again) • What would happen if we grouped the 3 × 4 first? What would the array look like? Draw the array. If we now multiply by 2, how many 3 × 4 arrays do we have? (two) Draw a second 3 × 4 array beside the first array. What is the product and how do you know? (24, because 12 + 12 = 24 or 2 × 12 = 24) • Are the products the same if we group the three numbers differently? Compare the two arrays with 24 circles. Where can you see the two groups of 12 from the second array in the first array? (e.g., There are two rows and 12 circles in each.) Does everyone see that? Circle one of the groups. • Look at the first array. We combined four 2 × 3 arrays. Where can you see these four 2 × 3 arrays in the second array? • The product is the same whether we group the 2 and 3 together first or whether we group the 3 and 4 together first. The only difference is that we have grouped them differently. Does the way in which the circles are grouped affect the total? • This is known as ‘the associative property.’ You don’t need to know the name, but it is important to know how it works and how it can help you. For example, if you are multiplying 6 × 5 × 2, is it easier to group the 6 and the 5 together first or the 5 and the 2? Turn and talk to your partner. • What did you find? (e.g., It was easier to multiply 5 × 2 because I know that fact well, and then I can multiply 10 × 6.) This is another mental math strategy that we can use when we are solving problems. Materials: Math Talk 3: a variety of concrete objects Math Focus: Investigating multiplying by 0 and 1 Teaching Tip Let’s Talk Integrate the math Select the prompts that best meet the needs of your students. talk moves (see page 7) throughout • Provide pairs with concrete objects such as counters or connecting cubes. Math Talks to maximize • Listen to this problem. There are five plates with two pieces of pizza on each student participation and active listening. plate. How many pieces of pizza are there? Work with your partner to create a concrete model for this problem. • How did you represent it? (e.g., I made five groups and put two counters in each.) How did you calculate the total? (e.g., I counted by 2s; I multiplied 5 × 2 = 10.) continued on next page Algebra (Equality and Inequality) 123

• Now represent this problem. There are two plates with five pieces of pizza on each plate. How many pieces of pizza are there? • How did you represent it? (e.g., I made two groups with five counters in each group.) How did you find the total? (e.g., I counted by 5s; I added 5 + 5; I multiplied 2 × 5 = 10.) • What is the same in both of these problems? (e.g., Both have 10 pieces of pizza.) What is different? (They are grouped differently.) • Represent this problem. There are seven plates and one piece of pizza on each. How many pieces of pizza are there? • What did you find? How can we represent this with a multiplication equation? (7 × 1 = 7) How many pieces would there be if there were 16 plates with one piece on each? 100 plates? What are the multiplication equations for each of these? • Why is the number of pieces of pizza equal to the number of plates in each case? (e.g., It is only one piece of pizza per plate in each case.) • What would be the pizza story if the multiplication is 1 × 7? (e.g., It would be one plate with seven pieces of pizza.) So, the story and the multiplication equation look different, but both the scenarios represent the same number of pieces of pizza. What can we say about multiplying by 1? (e.g., When you multiply a number by 1, the answer is always the same as the original number.) • Here is a new problem to represent. There are five plates and there are no pieces of pizza on any of them. How many pieces of pizza are there? • How did you model this problem? (e.g., five groups with nothing in them so there are zero pieces of pizza) How do we represent this as a multiplication equation? (5 × 0 = 0) • Why isn’t the answer ‘5’ because we have five plates? (e.g., The answer is the number of pieces of pizza and there are none.) • How can you represent 0 × 5? (e.g., no plates with five pieces on each, but there are no plates so there can’t be any pizza) • Is this true for any number multiplied by 0? Why? What rule can we make about multiplying by 0? We can add these rules to our anchor chart about properties. Further Practice • Reflecting in Math Journals: Have students describe in their Math Journals one example of multiplying by 1 and another example of multiplying by 0. They can use one of the examples shared during the lesson, or create their own. 124 Algebra and Data

11Lesson Using Coding Concepts to Investigate Properties of Operations Math Algebra Curriculum Expectations • C3.1 solve problems and create computational representations of Teacher mathematical situations by writing and executing code, including code that Look-Fors involves sequential, concurrent, and repeating events Previous Experience • C3.2 read and alter existing code, including code that involves sequential, with Concepts: Students have concurrent, and repeating events, and describe how changes to the code investigated equality affect the outcomes using concrete objects and equations. • C2.3 identify and use equivalent relationships for whole numbers up to 1000, in various contexts • Overall Expectation: C4 apply the process of mathematical modelling to represent, analyse, make predictions, and provide insight into real-life situations Number • B2.1 use the properties of operations, and the relationships between multiplication and division, to solve problems and check calculations Possible Learning Goals • Investigates the order of a series of instructions (code) involving addition and multiplication to discover how it affects the outcome • Begins to understand the differences in the power of the operations of addition and multiplication • Understands the difference between adding and multiplying, and accurately carries out operations with or without a calculator • Follows a code to create a pattern • Reorders the sequence of the code and records the resulting pattern • Compares the patterns created by reordering the sequence of a code • Determines whether the order affects the outcome • Creates a code that involves addition and multiplication Algebra (Equality and Inequality) 125

PMraotcheesmseast:ical About the caPonrmodbmpleruomnviincsoaglt,vinrineggp, ,rereseanstoinnign,g sscteorlanetnceetgicniteginstgo,orlesflaencdting, As students investigate the properties of the operations, they learn that, like addition and subtraction, multiplication and division are inversely related Math Vocabulary: and so ‘undo’ each other. While learning about multiplication and division, csoedqiunegn, cceo,ree,qouradl,err,epeating they also learn about the relationship between repeated addition and multiplication and between repeated subtraction and division. Through investigation, students can also learn how additive thinking (involved in addition and subtraction) is very different from multiplicative thinking (involved in multiplication and division). For example, while 2 + 2 = 4 and 2 × 2 = 4, repeatedly multiplying by 2 results in greater increases than does repeatedly adding 2, as it involves doubling each time. Students can also learn how the order of operations in a pattern core can affect how numbers grow in the resulting pattern, when one operation is addition and the other is multiplication. For example, for a growing pattern that starts at 2 and has a pattern core of + 2 × 2, the numbers in three repeats of the pattern are 4, 8, 10, 20, 22, 44. However, if the order of operations in the core is reversed to × 2 + 2, the numbers in three repeats of the pattern are 4, 6, 12, 14, 28, 30. While the numbers in the pattern are different, the growth reflects the power of the alternating operations. When comparing numbers of the same term, the number that increases by × 2 is always the larger (e.g., the 3rd term is 10 in the first pattern versus 12 in the second). The two patterns alternate as to which has the larger number with each increasing term (e.g., the 3rd term is 10 in the first pattern versus 12 in the second, then the 4th term is 20 in the first pattern versus 14 in the second). Although consecutive terms differ by either + 2 or × 2 in both patterns, the resulting number patterns are very different. Understanding all of these concepts may be beyond some students in grade three, but the important point is for them to realize that addition and multiplication make numbers grow in very different ways. About the Lesson In this lesson, students learn about coding concepts without the use of a computer. In Lesson 12 in the Patterning unit, students applied their understanding of coding concepts to discover that the order of multiplication and division instructions having the same numbers did not affect the outcome, because the two operations undo each other. In this lesson, students investigate how the order of a series of addition and multiplication instructions (code) affects the pattern or outcome, thereby uncovering the differences between additive and multiplicative thinking. Sequence of instructions and its effect on the outcome is a critical coding concept that grade three students are expected to understand. 126 Algebra and Data

Since students are writing and proving a conjecture in this lesson, it is a good opportunity to reinforce the Mathematical Modelling Process and how students move back and forth through the four components as they are developing, refining, and testing their model. There are suggestions throughout the lesson for how and when to highlight the modelling process; these will need to be adjusted to reflect how your students progress throughout the lesson. Materials: Minds On (20 minutes) BLM 11: Hundreds • Ask students how the core of a pattern is like the code that computer Chart, BLM 15: Number Lines, calculators, chart programmers give to computers to carry out a specific task and to achieve a paper desired outcome. Time: 60 minutes • Tell students that they are going to find out if the order of the elements in the core (code) affects the outcome. Ask students to explain this in their own words. (Understand the Problem) • Activate students’ prior knowledge by giving them a problem similar to one in the previous coding lesson (Lesson 12 of the Patterning unit). Tell students that a pattern starts at number 6 and has a core (code) of × 6, ÷ 2, ÷ 3. Have them figure out the pattern for three repeats of the core. • Have students alter the order of the elements in the core and discover what the pattern is. Give them copies of BLM 11: Hundreds Chart and BLM 15: Number Lines, or other materials or tools to use to solve the problem. (Assess the Situation) • Discuss students’ responses. Ask why all the sequences produced a repeating pattern or loop. (e.g., multiplication and division undo each other) Highlight how dividing by 2 and then dividing by 3 is the same as dividing by 6, except that the division is done in two parts. • Ask what repeating number patterns emerged from altering the order of the elements in the core (code). Some possibilities are shown below: – Start at 6: × 6, ÷ 2, ÷ 3 6, 36, 18, 6… – Start at 6: ÷ 2, × 6, ÷ 3 6, 3, 18, 6… – Start at 6: ÷ 2, ÷ 3, × 6 6, 3, 1, 6... • Discuss how the outcome is the same, but the number patterns are different. Working On It (20 minutes) • Show students a new core: + 2, × 2 and tell them the pattern starts at 2. Ask students whether they think the core will produce a repeating pattern (loop) Algebra (Equality and Inequality) 127

and whether changing the order of the elements in the code will alter the outcome. • Depending on students’ responses, write a conjecture. Explain that they need to test this, including with other numbers, in order to prove it can be a rule that works every time. Remind students that if one example doesn’t work, that disproves the rule. • Tell students they can solve the problem in any way they like, using any materials or tools they think will be helpful. They can also use numbers other than 2, as long as the two numbers in their code are the same (e.g., + 3, × 3). They can use calculators once the numbers become larger. (Create a Model) Differentiation • Some students may benefit from using a calculator for the entire investigation, so they can focus on results rather than struggling with calculations. • For students who need more of a challenge, give them a code with three elements, such as × 2, + 4, + 3. Assessment Opportunities Observations: This is a good opportunity to observe and listen. Note how well students follow instructions and make observations after each instruction is completed on both sides. Consolidation (20 minutes) • Meet as a class. Discuss students’ findings. Use + 2, × 2 as the first example since most students probably investigated this code. • Ask students what type of pattern the code created (growing). Ask why they think it did not create a repeating or shrinking pattern. List the patterns that students found: – Start at 2: + 2, × 2 2, 4, 8, 10, 20, 22, 44, 46, 92 – Start at 2: × 2, + 2 2, 4, 6, 12, 14, 28, 30, 60, 62 • Ask students to compare how the two patterns grow. Possible observations include: – Beyond the first two numbers (the first and second terms), when comparing the same term in each pattern, the odd-numbered terms (i.e., 1st term, 3rd term, 5th term,...) produced by the first code are always larger but the even-numbered terms (i.e., 2nd term, 4th term, 6th term,...) produced by the second code are always larger. – When comparing the same term in each pattern, the differences between the first and second terms in the two codes are the same (e.g., 0) and the differences between the third and fourth terms are the same (e.g., 2). This pattern continues for the differences in the pairs of terms moving forward (e.g., differences between the fifth and sixth terms are both 6). 128 Algebra and Data

– The differences between the number pairs keep getting larger as the pattern grows. – The difference between the numbers of the same term in the two patterns keeps getting larger as the patterns grow. • Have some students who used different numbers share their findings. Compare them to what was found with the first code. • Ask students why they think the different sequence in the codes produces different patterns. Explain that although the numbers in the sequence are different, they are growing in a similar pattern. Whichever term has grown by × 2 is always larger than the term that is increasing by + 2. While the two patterns are increasing in similar ways, the order of the sequence produces different number patterns. • Discuss the strategies students used. Ask whether they would do anything differently if they were to carry out a similar investigation. (Analyse and Assess the Model) Materials: Math Talk: “Magical Mix-Up” Math Focus: Investigating how the sequence of operations affects the outcome (page 17 in the Algebra and Data big book and Let’s Talk little books), counters, paper strips, chart paper Select the prompts that best meet the needs of your students. Teaching Tip • Show “Magical Mix-Up” (page 17 in the Algebra and Data big book). What Integrate the math do you see on this page? Who are magicians and what do they do? Have you talk moves (see ever seen a magician? Have you ever seen someone do a magic trick? What page 7) throughout was it like? Math Talks to maximize student participation • Why do many magicians have hats and wands? What often happens when and active listening. something goes into the magician’s hat? (e.g., Something disappears or something different comes out of the hat.) • Look at the order of the pictures. What story do they tell? (e.g., Some balls go into the hat, and we don’t know what comes out.) Would there be a different story if the pictures were in a different order? (e.g., Something goes into a hat and some balls come out.) So order matters, since it affects the outcome of the story. • Look at the instructions on the ripped pieces of paper. What do you think they are and why are they ripped? These are the magical instructions, and they are no longer in the correct order. The instructions are like the coding instructions we give to computers, so they carry out the commands and we get the desired outcome. • What do you notice is the same and different about all of the instructions? (e.g., They all have 2s, but they each have a different operation.) Do you think the order in which these instructions are carried out will affect the outcome? Why? continued on next page Algebra (Equality and Inequality) 129

Partner Investigation • You are going to investigate whether the order of the instructions can result in different outcomes. What happens with a code that creates a repeated event? (e.g., The outcome happens over and over again.) You are also going to investigate whether you can sequence an order that will result in a repeated event. • You are going to start with eight balls going into a hat and then perform each of the operations to find the outcome. You will then change the order of the instructions to see if you get a different outcome. • Print each of the instructions (parts of the code) on separate strips of paper so you can adjust the order easily. You can record the order of your instructions and the outcome. Try to find as many different orders and outcomes as possible. • While you are adjusting the orders, see if you can create a repeating event. Follow-Up Talk • W hat did you find? Let’s record some of the sequences that you used and what the outcomes are. Possibilities include: 8×2+2÷2−2=7 8÷2−2×2+2=6 • Which order produced a repeating event? How do you know that it is a repeating event? • W hy do you think the order of the operations affected the outcomes? Why is it important that we always check the order of our coding instructions to confirm that the code achieves our desired outcome? Building Social-Emotional Learning Skills: Self-Awareness and Sense of Identity: Have students reflect back on the algebra lessons and activities. Discuss how they would explain to another person what equality is and what variables are. Ask students what they think they understand well, and what may still be somewhat confusing. Make a list of the concepts that are not as clear, and ask them what kind of things they could do to practise (e.g., balancing two unbalanced quantities in different ways). Explain that over the next couple of weeks, the class will take 5−10 minutes a day to review the concepts on the list. Regularly refer to the list and ask whether any of the concepts are getting clearer for them and whether there are any points to add. In this way, students are taking part in their goal-setting. This also helps them to realize that learning doesn’t stop when the unit is done and that, with practice, they continually improve. 130 Algebra and Data