p106-193-Gr1-ON-Number-Unit2-add-subtract

Unit 2: Addition and Subtraction to 10 Lesson Content Page Introduction to the Addition and Subtraction Units 106 112 Addition and Subtraction to 10 114 119 1 Read Aloud: Math Fables: First Reading 125 126 2 Math Fables: Second Reading 128 131 3 and 4 Understanding the Equal Sign 134 140 3 Equality and Balance: Mass 143 149 4 Linking Equality and the Equal Sign 152 157 5 Representing Joining Problems 158 163 6 Investigating Strategies for Joining Problems 166 172 7 Addition: Commutative Property 175 178 8 Addition: Varying the Unknown 183 186 9 Representing Separating Problems 191 10 Strategies for Solving Separating Problems 11 to 13 Part-Part-Whole: Composing Quantities 11 Part-Part-Whole: Composing 5 12 Part-Part-Whole: Composing 6, 7, 8, and 9 13 Part-Part-Whole: Composing 10 14 Whole-Part-Part: Decomposing 10 15 Subtraction as ‘Think Addition’ 16 Compare Problems: Differences 17 Anchoring 5 and 10 Using Mental Strategies 18 Linking Addition and Subtraction 19 Reinforcement Activities

Introduction to the Addition and Subtraction Units There are two units dealing with addition and subtraction which emphasize conceptual understanding of the operations. Conceptual understanding is critical and should not be overlooked to get to ‘learning the facts.’ Lawson supports this statement by offering a quote from the National Research Council. “For students in grades K–2, learning to see the part-whole relations in addition and subtraction situations is one of their most important accomplishments in arithmetic” (Lawson, 2016, p. 56). Students will also have opportunities to develop and practise mental strategies for solving related problems by building on their conceptual understanding of the operations. The two units are: • Unit 2: Addition and Subtraction to 10 • Unit 4: Addition and Subtraction to 50 Note: Many of the lessons used in the earlier unit can be easily adapted to reinforce addition and subtraction to 50. About the As students count, compose, and decompose quantities, they learn about number relationships, which support understanding of the operations of addition and subtraction. Addition is often taught as meaning ‘put together,’ while subtraction is said to mean ‘take away.’ According to Van de Walle, these definitions are narrow, and can result in students having a limited understanding of the two operations and how they are related. Van de Walle explains that, “addition is used to name the whole when the parts of the whole are known [and] subtraction is used to name a part when the whole and the remaining part are known” (Van de Walle & Lovin, 2006, p. 66). Based on numerous research studies, he emphasizes that students can solve verbal problems by, “thinking through the structure of the problems rather than by identifying the type of action or key words” (Van de Walle & Lovin, 2006, p. 66). Materials Throughout the processes of addition and subtraction, students use various materials and tools to gain understanding and to visually ‘see’ the processes. Students may use a variety of concrete materials, such as connecting cubes, counters, or arithmetic racks. They may also use five and ten frames to help 106 Number and Financial Literacy

them anchor quantities of five and 10. When students can understand quantity represented as a distance, they may also use number lines and relational rods. Through these experiences, students can create images in their minds that can be later retrieved and manipulated when carrying out mental strategies. Structures of Addition and Subtraction Problems There are three basic structures of problems: • Join and separate problems • Part-part-whole problems • Compare problems Join and Separate Problems Join and separate problems both involve an action that takes place over time. Join problems have the beginning quantity increased, while separate problems have the beginning quantity decreased. Clements and Sarama highlight how the position of the unknown can affect the difficulty of the problems. For example, result unknown problems (5 + 4 = —) tend to be easier than change unknown problems (5 + — = 9), and start unknown problems tend to be the most difficult (— + 4 = 9) (Clements & Sarama, 2009, p. 62). The following table represents the different structures of problems. Action: Join Problems Example Position of Unknown Jesse has 4 balloons. He gets 3 more. How many balloons does he Result Unknown have? 4 + 3 = — Change Unknown Jesse has 4 balloons. He is given some more balloons. Now he has 7 balloons. How many balloons was he given? 4 + — = 7 Start Unknown Jesse has some balloons. He is given 3 more. Now he has 7 balloons. How many balloons did Jesse have at the start? — + 3 = 7 Action: Separate Problems Position of Unknown Example Result Unknown Sana has 7 cookies. She gives 4 away. How many cookies does she have now? 7 – 4 = — Change Unknown Sana has 7 cookies. She gives some away. Now she has 3 cookies. How many did she give away? 7 – — = 3 Start Unknown Sana has some cookies. She gives 4 away. Now she has 3 cookies. How many cookies did Sana have at the start? — – 4 = 3 Addition and Subtraction 107

Part-Part-Whole Problems There are two parts that comprise a whole, but no action takes place. The two parts remain separate as subsets. What is being highlighted is the relationship between the whole and its parts that remain separated. Unknown Example Whole Unknown There are 4 dogs and 3 cats in the yard. How many pets are in the yard? Part Unknown There are 7 pets in the yard. 4 are dogs and the others are cats. How many cats are there? There are 7 pets in the yard. 3 are cats and the others are dogs. How many dogs are there? Compare Problems The quantities of two sets are being compared, yet they are not subsets of each other, like in part-part-whole problems. Instead, the focus is on the difference between them. Unknown Example Difference Unknown Jesse has 4 cookies. Sana has 3 cookies. How many more cookies does Jesse have than Sana? (or How many fewer cookies does Sana have?) Compare Quantity Sana has 3 cookies. Jesse has 1 more cookie than Sana. How many cookies Unknown does Jesse have? (or Jesse has 4 cookies. Sana has 1 less than Jesse. How many cookies does Sana have?) Referent Unknown Jesse has 4 cookies. He has 1 more cookie than Sana. How many cookies does Sana have? (or Sana has 3 cookies. She has 1 less than Jesse. How many cookies does Jesse have?) Adding and Subtracting Strategies Many researchers and educators, including Thomas Carpenter and colleagues, Cathy Fosnot, Doug Clements, Alex Lawson, and John Van de Walle, have studied the strategies that students may use when solving addition and subtraction problems. Following is a synthesis of many of their ideas and findings. More in-depth analysis is available in their books listed in the Reference section (see pages 472–473). As young children solve problems that are related to addition and subtraction, they intuitively directly model the situation and apply many of their counting skills. With experience, they develop a variety of strategies which become increasingly efficient and more abstract. It is important to note that students will progress in different ways and may or may not engage in all of the strategies. Alex Lawson points out that transition from one strategy to another is not rigid and students may revert to previous and perhaps less efficient strategies, depending on the types of problems that they encounter (Lawson, 2016, p. 18). Carpenter’s research indicates that students generally progress from direct modelling to using various counting strategies to attaining more automatic recall of facts by understanding number relationships, and then to 108 Number and Financial Literacy

using known facts to derive other calculations (Carpenter, Fenema, Loef Franke, & Levi, 1999, p. 26). Lawson’s studies reveal that students also use strategies, such as decomposing quantities into groups of 5 and 10 for easier calculation (Lawson, 2016, p. 21). The strategies that students choose are often dependent on the type or structure of the problem and the size of the numbers. For example, students may revert to an earlier strategy as the numbers get larger. Your goal is to provide students with opportunities to develop more effective strategies over time. Direct Modelling Students often begin directly modelling problems by acting them out, using concrete materials, or creating drawings to represent the groups and then recreating the action that takes place. The following problems will be used to illustrate students’ thinking: • Pear Problem: There are 3 pears in a bowl. Jen adds 4 more pears. How many pears are in the bowl now? • Butterfly Problem: There are 7 butterflies in the garden. 3 butterflies fly away. How many butterflies are left? Students use concrete materials to represent the objects and count three times. Pear Problem: • Count to create a group of 3. • Count to create a group of 4. • Mimic the joining action by pushing sets together. • Count the new group as 1, 2, 3, 4, 5, 6, 7. Butterfly Problem: • Count to create a group of 7 objects. • Remove 3 objects, counting 1, 2, 3. • Count the remaining objects 1, 2, 3, 4. In both cases, the concrete materials directly represent the objects in the problem. Lawson states that the use of concrete objects “decreases the amount of information that needs to be held in one’s head” (Lawson, 2016, p. 19). Counting Strategies A major step occurs when a child no longer sees the need to physically model the situation, but counts in various ways to find a solution. Counting On and Back Pear Problem: • Count on from 3 as 4, 5, 6, 7, raising a finger with each count to track the addition of 4 pears. • Count on from the larger set of 4 as 5, 6, 7, raising a finger with each count to track the smaller set of pears. Addition and Subtraction 109

Butterfly Problem: • Say 7, and then count back 6, 5, 4, raising a finger with each count to track the 3 butterflies that fly away. In the counting on strategy, the child counts on from 4 as 5, 6, 7, raising a finger each time. As Lawson points out, raising a finger with each count is tracking the number sequence (Lawson, 2016, p. 20). This strategy is more abstract than directly modelling the problem and more efficient. It takes time for students to transition from using direct modelling to counting strategies, and in many cases, they will use one or the other, depending on the problem they are solving. Anchoring Five and Ten Students will often decompose quantities into groups of 5. For example, 4 + 7 can be seen as 5 + 6. As they mentally ‘move’ one unit from the 7 to the 4 to create 5, they are creating equivalence through compensation. As Fosnot explains, if you lose one from one number but gain it in the other number, the total stays the same (Fosnot, 2007, p. 6). Students may also decompose 4 + 7 into 5 + 5 + 1, and then recompose to make 10 + 1, thereby composing a ‘friendly’ group of 10. Composing groups of 10 is important since it aligns with our base ten number system and the concept of place value. For example, 9 + 7 can be seen as 10 + 6, or 1 ten and 6 ones. Once students understand the inverse relationship of addition and subtraction, they can ‘think addition’ when confronted with a subtraction problem. For example, they may see 11 – 5 as 5 + — = 11, or they may count on from 5 to 10, and then add one more to get to 11. As Lawson highlights, students are “thinking of the difference, not as the result of removal, but as the distance” between the two numbers (Lawson, 2016, p. 21). Automatic Recall of Some Facts and Using Those Facts to Learn Derived Facts Through experience, students can recall certain facts, such as doubles or adding 1 more or less. They can then use these facts to figure out other facts by understanding the relationships between the numbers. For example, they may recall that 4 + 4 = 8 and use this to know that 4 + 5 will be 1 more than 8 in the counting sequence (Carpenter et al., 1999, p. 30). Gaining Automaticity with Math Facts Students in grade one work on recalling addition facts and related subtraction facts up to 10. Cathy Fosnot states, “understanding what it means to add and subtract is necessary before facts can become automatic, but understanding does not necessarily transfer to automaticity” (Fosnot & Dolk, 2001a, page 97). Alex Lawson adds that this ‘automaticity’ evolves over time by “working with different strategies and the construction of key ideas that can be applied across calculations” (Lawson, 2016, p. 22). This is more effective than memorizing facts in isolation since the major focus is on the relationships among numbers and the operations. This takes time and evolves over several years. According 110 Number and Financial Literacy

to the math curriculum, addition and subtraction facts should be mastered by the end of grade three (Ontario Ministry of Education, 2020, p. 33). Students in grade one therefore have time to gain automaticity. Caution About Timed Tests and Activities A traditional approach has been to time students while they solve math calculations in order to assess their recall. While we want students to gain proficiency with these calculations, we should question the use of timed tests and activities based on the research. Jo Boaler writes about how “timed tests cause the early onset of math anxiety for students” and is exacerbated over time, leading to “low achievement, math avoidance, and negative experiences of math throughout life” (Boaler, 2014, p. 469). Sian Beilock’s research indicates that the stress actually blocks working memory, restricting people’s ability to recall familiar facts and details (Boaler, 2014, p. 370). Not only does this affect math achievement, but also self-beliefs. When students underachieve because they cannot access their working memory, they question their own abilities, lose self-confidence, and develop math anxiety. Other Approaches Boaler states that there are many ways to introduce and reinforce mental strategies without the pressure of being timed, such as through discussions about how numbers are related, and applying strategies in games and activities. She also notes that students best internalize math facts when they engage in mentally solving number problems on a daily basis (Boaler, 2014, p. 47). The Development of Mental Strategies In grade one, students are expected to “use mental math strategies, including estimating, to add and subtract whole number that add up to no more than 20” (Ontario Ministry of Education, 2020). It is also important that students explain their strategies, as this reflects their understanding of numbers and their relationships. Throughout these units, students apply their spatial reasoning skills and use visual representations of quantities and how they are organized in order to create effective mental images. These can be translated into numbers and equations, making the connections between the two representations evident. The intent of the lessons in the Addition and Subtraction units is to develop conceptual understanding of the operations of addition and subtraction. Students’ experiences with concrete materials and tools help them form the visual images that allow them to develop mental strategies. Addition and Subtraction 111

Addition and Subtraction to 10 PMraotcheesmseast:ical About the tccarPooeonornpmodlnsrbempelasercunoemtndvinniitncsginaog, gtl,svi,rneienrgleefg,lace,tsciontingnign,g strategies Students in kindergarten informally “investigate addition and subtraction in everyday experiences and routines through the use of Math Vocabulary: modelling strategies and manipulatives” (Ontario Ministry of ebtesshqsacdqicuueiglogeuoabanmnscamntl,aro,,tcpeaibmnmteooqaciodnnsetupkt,,,aeeaoaenma,tlsisaqodoesi,wutdnni,qgabu,asnulpasy,a,,,lluanmsnti,ictnpeyu,ldus,s Education, 2016b, page 225). In grade one, students build on these informal experiences to gain a more formal understanding of addition and subtraction. Students are exposed to a variety of problem structures, as well as problems that have the unknown in different positions, requiring them to interpret the operations through the context. They apply their abilities to compose and decompose quantities in order to anchor benchmark numbers, such as 5 and 10. As they build equations to match the context and operations in the stories, they learn about the symbols, such as plus and minus signs, and gain an understanding of equality as a balance and the equal sign as meaning ‘the same as.’ Throughout the investigations, students discover the relationship between the operations of addition and subtraction. 112 Number and Financial Literacy

Lesson Topic Page 1 Read Aloud: Math Fables: First Reading 114 119 2 Math Fables: Second Reading 125 126 3 and 4 Understanding the Equal Sign 128 131 3 Equality and Balance: Mass 134 140 4 Linking Equality and the Equal Sign 143 149 5 Representing Joining Problems 152 157 6 Investigating Strategies for Joining Problems 158 163 7 Addition: Commutative Property 166 172 8 Addition: Varying the Unknown 175 178 9 Representing Separating Problems 183 186 10 Strategies for Solving Separating Problems 191 11 to 13 Part-Part-Whole: Composing Quantities 11 Part-Part-Whole: Composing 5 12 Part-Part-Whole: Composing 6, 7, 8, and 9 13 Part-Part-Whole: Composing 10 14 Whole-Part-Part: Decomposing 10 15 Subtraction as ‘Think Addition’ 16 Compare Problems: Differences 17 Anchoring 5 and 10 Using Mental Strategies 18 Linking Addition and Subtraction 19 Reinforcement Activities Addition and Subtraction to 10 113

1Lesson Math Fables: First Reading Introduction to the Read Aloud The Read Aloud text introduces math concepts in a meaningful context that allows students to make connections to their everyday lives. In the first reading, students apply their literacy strategies, such as making connections, inferring, predicting and synthesizing information, to understand the animal fables told throughout the book and the life lessons that can be learned from them. The lesson can be further extended by hearing animal stories from other sources, such as from Indigenous cultures, and discussing any common messages that are conveyed. Students can also discuss the needs and characteristics of animals in the book to link to the science curriculum. During the second reading, students are mathematicians and apply the mathematical processes to discover and explore the math concepts embedded in the context of the story and its rich illustrations. Both readings are valuable for assessing where students are, what some of their misconceptions might be, what concepts need greater emphasis, and what differentiation may be necessary. Since the context of each fable stands alone, the first reading can be broken up and read over two or three days, allowing more discussion about the lessons embedded in the fables. Similarly, the second reading can be presented over a couple of days, following up each section with a partner investigation to further explore the math. Language Oral Communication Curriculum Expectations • 1.3 identify a few listening comprehension strategies and use them before, during, and after listening in order to understand and clarify the meaning of oral texts, initially with support and direction • 1 .4 demonstrate an understanding of the information and ideas in oral texts by retelling the story or restating the information, including the main idea • 1.5 use stated and implied information and ideas in oral texts, initially with support and direction, to make simple inferences and reasonable predictions • 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 Science and Needs and Characteristics of Living Things Technology Curriculum • 2.3 investigate and compare the physical characteristics of a variety of plants Expectations and animals, including humans Visual Literacy • Numerals appear in larger print and different colours 114 Number and Financial Literacy

Materials: Assessment Opportunities Written by Greg Tang Observations: Note each student’s ability to: Illustrated by Heather – Make predictions and simple inferences, and demonstrate Cahoon understanding by responding to questions and class discussion Text Type: Fiction: Fable — Poem – Connect and ask questions to uncover a moral to a story Time: 2 0–30 minutes – Use illustrations to form ideas about how animals adapt to their environment Read Aloud: Math Fables Summary: The text is made up of several short fables, each dealing with a group of animals that is one larger than the group on the previous page. As different animals live and adapt to the challenges in their environments, a moral emerges, which young children can relate to their own lives and to their interactions with others. NOTE: While you may not choose to have in-depth discussions about each story, probing questions for all of them are outlined below. Select the prompts that best suit the needs and interests of your students. Before Reading Inferring/predicting/ Activating and Building On Prior Knowledge connecting Inferring • Show the front cover of the book. Read the title and the names of the author and illustrator. Read the words at the top of the cover, “Lessons That Count,” and ask students what a fable might be. Add to students’ ideas so they understand that a fable is a short story, usually with animals as the characters, that can teach us a lesson about our lives. Ask whether they can think of a story that has taught them a lesson. • Turn to the page entitled “Going Nuts.” Ask how these animals are different from real animals (e.g., they are wearing scarves and mittens, they are smiling and dancing). Ask why they think the author wanted the animals to have human characteristics (e.g., so children can relate to them better). Ask what parts of the illustrations are realistic (e.g., the environment, the general appearance of the animals). • Setting a Purpose: Say, “We have some ideas about the unique characteristics of the animals in this book. Let’s read to find out what we can learn from these animals as they explore their environments.” During Reading Inferring/predicting “Dinner Guest” • Before reading the page, ask students what animals they see and what is happening in the illustrations. Ask what action may take place next. Read the text on the pages. Ask how the words ‘who would come to dinner now’ can have two meanings (e.g., come to dinner as a guest or to be eaten). Addition and Subtraction to 10 115

Connecting • Ask students what lesson can be learned from this story. Ask them to think Inferring about a time when they have had to be patient. Discuss how learning something new often takes patience and time. • Ask students how the animals that will get caught in the web are going to feel and why. Inferring/predicting “Trying Times” Evaluating the author’s craft/ • Ask students what they think is happening in the illustrations. Ask how they predicting Connecting think the two baby birds in the nest feel and why. Read the text on the first page of the spread. • Read the first stanza on the second page of the spread, but leave out the last word. Have students predict what the word might be. Ask what helped them figure this out (e.g., the rhyming pattern, the illustration). Read the second stanza and leave out the last word. Have students predict what the word is. • Ask what the lesson of this story is. Ask when they have had to try and try again to get better at something. Ask how this made them feel when they finally succeeded. Connecting/ “Family Affair” visual literacy • Ask students what they know about turtles. Ask how the pictures help them Connecting understand more about turtles and the type of environment they need in order to survive. Ask what features turtles have to protect them from predators. Read the text. • Ask students what the lesson is in the story. Ask when they have relied upon their family to help them. Inferring “Going Nuts” Solving word meaning from • Ask students what time of year it is and how they know. Ask why the squirrels context Inferring might be so happy. Read the first spread and then ask what problem the squirrels face. Connecting • Turn the page and read the first page of the spread. Ask what they think the words ‘mound’ and ‘stashes’ mean, and why they think so. • Ask why the squirrels bury their acorns. Ask how this helps them prepare for the upcoming winter. Ask how they think squirrels survive in the winter. • Ask what lesson can be learned from the squirrels. Ask students whether they have ever prepared for something ahead of time so they would be ready. Inferring/ “Midnight Snack” connecting • Ask what animals are in the story and what they are doing. Ask students if they have had any experiences with raccoons. Ask why the raccoons are active at night. Ask what other animals in the illustration are awake. Explain that owls and raccoons are nocturnal, meaning they are active at night and sleep during the day. Read the first spread of text. Ask what features the raccoons have to help them get into the garbage can. Ask whether raccoons are friends or enemies of humans and why. 116 Number and Financial Literacy

• Read the second spread of the story. Ask what the raccoons are thankful for. Ask what the lesson of the story is. Ask students what they are thankful for. Visual literacy “Tool’s Gold” Inferring • H ave students study the illustrations and describe details of the otter’s Connecting environment. Ask what is not real in the pictures (e.g., otters wearing flippers, otters smiling). Ask what features otters have that help them thrive in their environment. Ask what impact humans may have on this environment. Read the text on the first spread. • Turn to the second spread, and draw attention to the first picture. Ask why some of the otters look worried and what problem they may have (e.g., they can’t open the shells). Read the second spread. Ask how the otters solved their problem. • Ask what lesson the otters learned. Ask students how they use tools and why they are helpful. Inferring/connecting/ “Gone With the Wind” predicting • Ask students what animals are shown and to what category they belong Visual literacy/ evaluating (insects). Ask what some of the features of insects are. Ask what they know about the life cycle of a butterfly. Read the text on the first spread. Ask what Synthesizing/ problem the butterflies have. Ask what they might do to solve their problem. connecting • Turn to the second spread. Ask students why the illustrator shows the butterflies flying during the day and during the night. Read the text. Explain that the butterflies are migrating. • Ask what lesson the butterflies learned. Ask if they have ever put off something that should have been done right away. Ask what they think the word ‘procrastinate’ means. Inferring/connecting/ “Profile in Courage” predicting • Read the title. Ask what ‘courage’ means and when they have had to have Evaluating the author’s craft Inferring/predicting courage. Ask what animals are featured on these pages. Ask students what they Inferring know about crabs and any experiences they have had. Ask what is not real about the crabs shown. Using text features to support word meaning • Read the text on the first page of the first spread, leaving out the final word. Synthesizing/ Have students predict the word and explain what helped them figure it out. connecting • Read the rest of the first spread. Ask what problem the crabs have. Have them predict where the crabs are going and why. • Turn to the second spread. Ask what they think is happening in the first two illustrations and why they think so. • Read the text on the second spread. Ask what they think the word ‘cautiously’ means. • Ask what lessons the crabs learned. Ask students when having friends has been helpful to them. Addition and Subtraction to 10 117

Inferring/connecting “Antics” Connecting • Ask what animals are in the illustration and to what category they belong Inferring/predicting (insects). Ask which features make them insects. Ask if they have had any Synthesizing/ experiences with ants. connecting • Ask how the people who made the picnic would feel if they saw the ants coming. Ask how this is different from the perspective of the ants. • Read the text on the first spread. Ask what problem the ants have and what they think the ants will do to solve it. • Turn to the second spread and read the text. Ask what lesson the ants learned. Ask what they think the word ‘cooperate’ means. Ask when they have found cooperating to be helpful. Inferring/connecting/ “River Sticks” predicting • Ask students what animals are shown and what they know about them. Ask Inferring what the animals are doing in the illustrations. Ask them to predict why the Using text features to beavers are gathering sticks. support word meaning • Read the text on the first spread. Ask what problem the beavers have. Ask what Inferring they think the word ‘canal’ means. Synthesizing/ connecting/inferring • Turn to the second spread. Ask students how the illustrations help them understand what ‘canal’ means. • Ask what features the beavers have to help them with their work. Ask what parts of the story are not real. • Ask what lesson the beavers learned. Ask students how working hard with others has been beneficial to them. Ask what they think ‘domestic engineer’ might mean. After Reading Analysing • Ask why the author and illustrator gave the animals human characteristics Connecting and how this helped them relate to what the animals are experiencing. • Invite Indigenous members of your community to visit and share their stories about animals and the lessons that humans can learn from them. Stories can be presented orally, in written stories, or through artwork and songs. • Your visitors can also share the Indigenous names of local animals, the behavioural features that help the animals adapt to their environment, and their significance to the local culture, both in the past and in the present. Building Social-Emotional Learning Skills: Stress Management and Coping: Ask students which fable they connect with the most and why. Discuss how many of the lessons in the fables can be applied to situations in which they feel stress or overwhelmed in math class. You may decide to pick one fable that has a particular message you want to reinforce. By building a repertoire of coping strategies, students develop resilience when facing challenges. 118 Number and Financial Literacy

2Lesson M ath Fables: Second Reading Math Number Curriculum Expectations • B2.1 use the properties of addition and subtraction, and the relationship Teacher between addition and subtraction, to solve problems and check calculations Look-Fors • B2.2 recall and demonstrate addition facts for numbers up to 10, and related subtraction facts • B1.3 compare and order whole numbers up to and including 50, in various contexts Possible Learning Goal • Explores ways to make 10 within the context of a story, using illustrations and text • C ounts objects in a set only once using a tracking system • C ounts sets of objects to 10, starting with different objects and reaching the same total • F lexibly decomposes quantities to 10 using concrete materials and visualization • P redicts one more or one less than a given quantity About the Greg Tang has stated that his goal for writing this book was to present numbers in a way that will make math easier for all children. The text initially introduces numbers in the traditional manner of counting quantities, which reinforces counting principles such as one-to-one correspondence, conservation, stable order, and cardinality. The progressive increase in the number of animals from story to story also reinforces the concept of hierarchical inclusion—that all previous numbers are nested within a number and that numbers grow by one with each count. The story then challenges readers to perceive these quantities decomposed in a variety of ways by creating different groupings. This is important since being able to flexibly think of quantities made up of smaller groups lays the foundation for our base ten number system. As students form groups, they are unitizing. For example, they cannot only see the 10 individual units in a group, they can also simultaneously see the 10 units as one group. Unitizing also reinforces students’ ability to skip-count using equal subgroups of a larger set. continued on next page Addition and Subtraction to 10 119

The students are also interpreting the grouping of animals within a context, which allows them to change the story but keep same-sized groups. For example, they can describe groups of two and three as two animals walking and three animals swimming or as three animals walking and two animals swimming. This gets students to think about the groupings differently, yet they still produce the same total of animals. Decomposing quantities to 10 in various ways will help students develop mental images they can use during their learning of addition and subtraction facts to 10. About the Lesson Now that students understand and have discussed the fables in the previous lesson, they can deepen their mathematical understanding by investigating the embedded math concepts. Since the text is rich in mathematical ideas, it is suggested that the reading be completed over two days, each session being followed with a student investigation based on what was discussed. Since there are more prompts than are feasible for these lessons, it is important to select the ones that best meet your students’ needs and interests. Materials: Assessment Opportunities Math Fables, Observations: Note each student’s ability to: concrete materials (optional) • Count, incorporating one-to-one correspondence, conservation, stable Time: 15–20 minutes (per day for two days) order, and cardinality • Decompose and recompose a quantity, realizing that the total remains the same • Create a different story for the same grouping of animals (e.g., two are playing and three are working, or three are playing and two are working) NOTE: Prompts are provided for reading all the pages over two days. Alternatively, you may decide to go into more depth with some of the pages; adjust the number of days as needed. Before Reading Reasoning and Activating and Building On Prior Math Knowledge proving • Ask students what they notice about the number of animals in each fable, progressing through the book. (e.g., The first page has 1 animal and each story progressively has 1 more animal, up to 10.) • Setting a Purpose: Tell students that as mathematicians, they will focus on exploring the math in the stories. 120 Number and Financial Literacy

During Reading Reasoning and Day 1 proving “Dinner Guest” • Read the spread. Ask students what ‘1’ means. Point out the mosquito and ask whether it is still ‘1’ even though it is smaller than the spider. Ask how many spiders would be left if this one walked away. (e.g., 0) Ask what ‘zero’ means. Problem solving “Trying Times” • Ask how much larger 2 is than 1. Read the text on the spread. Ask if there is any way to group the 2 birds, other than 1 and 1. Ask what other things come in 2s. Explain that items that come in 2s are often referred to as pairs. Problem solving “Family Affair” • Ask how much larger 3 is than 1, and 3 is than 2. Ask what they notice about the number of animals per illustration. (There are always 3.) Ask if there is another way the turtles might have been split up. (e.g., 2 turtles flip into the water and 1 turtle is left on shore; or 1 turtle went into a hole, 1 went into the water, and 1 stayed on the shore) Ask why the total remains the same. (e.g., no turtles are disappearing and no extra turtles are coming) Read the text. Problem solving/connecting/ “Going Nuts” reflecting • Read the first spread. Ask if there is another way to group the squirrels, having 3 in one group and 1 in the other. (e.g., 1 squirrel is frightened in the tree and the other 3 are telling it to get to work.) Ask if there is another way that they could be grouped. (e.g., 2 with mittens and 2 with scarves) • Turn to the next page and read the spread. Ask how the story could be changed yet still reflect two groups of 2, totalling 4. (e.g., It might be different squirrels doing different jobs, but there would still be 2 squirrels gathering and 2 squirrels burying acorns.) • Review these pages and then go back to the pages that show totals of 2 and 3 animals. Ask if there is a pattern in how the animals are grouped throughout the stories. (e.g., the total number of animals are together at the beginning, separated into groups in the middle, and then together again at the end) Problem solving/ “Midnight Snack” communicating/ • Show only the first page of this story, which shows the 5 raccoons walking representing toward the garbage can, and read the text. Ask how the raccoons might be grouped by the way they look. (e.g., 4 adults and 1 baby; 3 on two feet and 2 on four feet) Draw attention to the other side of the spread and ask students to tell a story about the raccoons and how they are separated into two groups. (e.g., 2 are tipping the garbage can and 3 are cheering them on) Read the rest of the first spread. Addition and Subtraction to 10 121

• F urther Investigation: Have students work in pairs. Have them find other ways that the 5 raccoons could be split into two groups. (e.g., one group of 2 holding hands and one group of 3 that are not) Students can use concrete materials or create drawings of the possibilities they find. • Further Talk: Discuss the groupings that the students found. Ask how 2 tipping and 3 cheering is the same and different from 3 tipping and 2 cheering. Ask how they could make a story with 5 in one group and 0 in the other. Ask what 0 represents. Reasoning and proving/ Day 2 problem solving “Tool’s Gold” • Show the first page of the first spread and ask students how they could group the animals. (e.g., by the colour of their flippers; 2 above water and 4 below water) Read the spread. Ask how else the 6 otters could be grouped. (e.g., groups of 5 and 1; two groups of 3) • Show the second spread. Ask what action happened to have the first grouping change into the second grouping. (e.g., 2 otters from the group of 5 joined the 1 otter) Ask if there are any other ways to group the 6 otters. Problem solving “Gone With the Wind” • Show the first spread. Ask students if they can figure out how many butterflies there are without counting them one by one. (e.g., see a group of 3 and a group of 4, and then counting on from 3) Ask how the 7 butterflies on the first page could be grouped, according to the illustration. (e.g., a group on the top branch and a group on the bottom branch; a group with open wings and a group with closed wings) Ask how one of the groupings changed from the first to the second picture. (e.g., now there are 5 on or near the top branch and 2 on the bottom branch) Read the text. Ask students to predict how else the 7 butterflies could be grouped. • Turn to the second spread. Ask how the groupings in the picture of the butterflies during the day could be transformed into the groupings in the picture of the butterflies at night. (e.g., 2 butterflies from the group of 6 join the leader) Ask how they could group the butterflies in the last picture. Read the text. Reasoning and proving/ “Profile in Courage” problem solving • Show the first spread. Ask students whether they can figure out the number of crabs without counting them one by one. (e.g., seeing a group of 5 and a group of 3, and then counting on from 5) Read the text on the spread. Ask how else the crabs could be grouped. Have students explain their reasoning. • Turn to the second spread and read the text. Ask if there is another story that could be created to depict one of the groupings. In the last picture, ask if there is another way to figure out how many crabs are under the one on top without counting by 1s. (e.g., 1 less than 8 is 7) 122 Number and Financial Literacy

Problem solving/ “Antics” communicating/ • Show the first spread and draw attention to the ants on the left. Have representing students visualize groupings for the ants so they are easier to count. (e.g., two groups of 4 and 1 more) Draw attention to the bottom picture on the right side of the spread. Ask how they can figure out how many ants are on the picnic blanket without counting them one by one. (e.g., take the 1 ant on the cheese from 9, and that leaves 8 ants) Read the text. Ask what other groupings could be made from the 9 ants. • Show the second spread and verify that they found all the combinations. Ask students whether they can divide the ants into two equal groups, and to explain their reasoning. Ask if the ants could be divided into three equal groups. Have them prove it. Communicating/ “River Sticks” representing/connecting/ • Show the first spread. Ask students how they can figure out the total number reflecting of beavers by looking at the illustration on the left. (e.g., they may see 6 and 4, or 7 and 3) Ask what groupings they see on the right side. Ask what action might have taken place to go from 7 beavers and 3 beavers to 9 beavers and 1 beaver. (e.g., 2 beavers left the group of 3 and joined the group of 7) Read the text. Ask what other groupings can be made with 10 beavers. • Show the second spread and verify if students found all of the combinations. Ask why there can be equal groups made from 10 but not from 9. Read the text. • Further Investigation: Now that students have explored different groupings for the numbers 1 to 10, have them investigate which numbers can be divided into two equal groups. They can use concrete materials or create drawings to figure this out. • Further Talk: Ask why certain numbers can be divided into two equal groups and other numbers cannot. Ask if they see a pattern. (e.g., every second number) As students slowly count to 10, drop a cube into an egg carton with every spoken number (one in the top row, then one in the bottom row), and ask what pattern they notice in the visual representation and in the order of the numbers (e.g., every second number is paired up). After Reading Communicating/ • Ask about students’ strategies for grouping various numbers of animals. Did representing/connecting/ they use trial and error, or did they have a systematic way of finding the reflecting groups? Ask how they knew they had found all of the groups. • Ask how the stories can differ. (e.g., One story had 2 animals swimming and 3 animals walking, and the other had 3 animals swimming and 2 animals walking, but they are both combined groups of 2 and 3 to make 5.) Addition and Subtraction to 10 123

• Building Social-Emotional Skills: Critical and Creative Thinking: Ask students what they wonder about when creating groups of 10. Ask how making 10 in different ways can help them make bigger numbers, such as 11 or 20, in various ways. Tell students that mathematicians are curious about math because they don’t know everything there is to know about numbers. They enjoy discovering new information by experimenting with numbers. Remind students that they may not know certain things about larger numbers YET, but they have lots of time in the years to come. Further Practice • Independent Problem Solving in Math Journals: Students can draw a group of at least 2 to 10 animals, divide their set into different groupings, and record the numbers beside their illustrations to show the quantities. • Reflecting in Math Journals: Verbally pose one of the following prompts: − Using pictures, numbers, and/or words, show the math that is on your favourite page. − Using pictures, numbers, and/or words, show where you see math and numbers in your life. Scribe any further ideas that students may verbally express. 124 Number and Financial Literacy

and3 4Lessons Understanding the Equal Sign Math Number Curriculum Expectations • B1.1 read and represent whole numbers up to and including 50, and Previous Experience describe various ways they are used in everyday life with Concepts: It could be beneficial if students Algebra have worked with pan balances before, but not • C2.2 determine whether given pairs of addition and subtraction expressions necessary. are equivalent or not PMraotcheesmseast:ical crrPeeorapmosrbemolsenueminnngitcsinaoagtlnvi,ndinrgepgfr,loevcitningg, , A bout the According to an extensive study carried out by Falkner, Levi, and Carpenter, less than 10% of students in any grade from one through six could give the correct response to the question 8 + 4 = — + 5. With greater probing, the researchers discovered that most elementary students have serious misconceptions about what the equal sign means (Carpenter et al., 2003, p. 9). Many interpret it as meaning ‘the answer is.’ According to the curriculum, grade one students are expected to “determine whether given pairs of addition and subtraction expressions are equivalent or not” (Ontario Ministry of Education, 2020). The concept of equality can initially be done using a pan balance so students can visually see how equal quantities balance each other. The research supports the value in taking considerable time to ensure students understand equality as a balance so later misconceptions can be avoided. About the Lessons The following lessons allow students to investigate ‘balanced’ and ‘not balanced’ using mass on a pan balance. Through their investigations, they may use the term ‘weight’ since it is more familiar to them. While mass and weight are different concepts, describing it as weight will be fine at this point in time. The lesson is more about students being able to experience ‘balance’ in a concrete manner. The vocabulary ‘equal’ and ‘not equal’ are introduced. Addition and Subtraction to 10 125

3Lesson Equality and Balance: Mass Teacher Possible Learning Goals Look-Fors • Investigates ‘balance’ of mass using concrete materials Math Vocabulary: • Explains or shows that ‘equal’ means ‘balanced’ by giving examples and ebpaqaasuln,aamnbl,caanelsaodsnt,,cenleeqov,utetabhl,lae, lasnacmeed, even non-examples Materials: • Creates balance on a pan balance using various objects pan balances, items • R ecognizes imbalance on a pan balance from around the • M akes reasonable predictions about what might balance an object room, chart paper, • R efines ideas about what might balance an object after several trials markers • E xplains what ‘balance’ and ‘equal’ mean Time: 50 minutes Minds On (15 minutes) • Show a pan balance with the two sides level. Ask students what they notice about the two sides. Highlight the vocabulary that students might use such as ‘even’ and ‘level.’ Introduce the word ‘balanced’ and explain how it is similar to their vocabulary. • Ask students to visualize what will happen if an object (e.g., an eraser) is put on one side. Have students turn and talk with a partner. As a class, discuss their predictions and reasons for them. Put the item on the one side and discuss what happens in relation to their predictions. Ask what they think might happen if another eraser is added to the same side. • Have students discuss with a partner what might be put on the other side to make the two sides balanced again. As a class, try some of their suggestions. Frequently describe the scale in terms of ‘balanced’ and ‘not balanced.’ • Put something very heavy, but small, on one side and show students a collection of smaller objects (e.g., erasers). Have them discuss with a partner how many of the smaller objects need to be put on the other side to create ‘balance.’ An accurate number is not expected, but just the idea that there would be a lot of the smaller objects needed. This helps students gain a general sense of larger numbers when they are estimating. Working On It (20 minutes) • H ave students work in pairs or small groups, depending on how many balances are available in the class. Have them experiment with creating balance. They can select items from around the room. Have them predict what might create balance before actually putting items on the other side. They can record objects that created balance by drawing pictures on chart paper. 126 Number and Financial Literacy

Differentiation • For ELLs, using several terms for one concept, such as ‘level,’ ‘even,’ or ‘balanced,’ can cause confusion. While students are working in small groups, connect the terms. For example, say, “You said that the two sides are even. That means they are balanced. If they are not even, they are not balanced.” Assessment Opportunities Observations: Pay attention to how students create balance. Do they randomly try objects, or do they selectively choose objects based on some kind of reasoning? Do they refine their choices after trying various objects? Are they predicting before putting objects on the balance? Conversations: For students who seem to be working randomly, ask them to visualize and predict what would make the two sides even. Before they place the objects on the scale, ask if their prediction makes sense and why they think so. Consolidation (15 minutes) • Have one pair or group identify an item that they tried to balance. Have the rest of the class visualize and predict what might balance the item. The group members can then reveal their findings. Have some of the other groups present their findings in a similar manner. • Ask students what the pan balance looks like when it is balanced and when it is not balanced. Have them explain the term ‘balance’ in their own words. • Ask students what creates the balance. (e.g., the same mass on each side) Tell students that the masses on each side are equal. Ask what they think ‘equal’ means. (e.g., ‘the same as,’ or ‘balanced’) Use the terms ‘equal’ and ‘not equal’ as you model one example on the pan balance. • Co-create with students an anchor chart with one picture that shows ‘balanced’ and one that shows ‘not balanced.’ Include the labels ‘balanced,’ ‘not balanced,’ ‘equal,’ and ‘not equal.’ Add these terms to the Math Word Wall. Further Practice • Independent Activity in Math Journals: Have students draw one picture that depicts the balance they found between objects, and one picture that depicts objects that are not balanced. They can label their drawings as ‘balanced’ or ‘equal’ and ‘not balanced’ or ‘not equal’ by using the anchor chart as a reference. Addition and Subtraction to 10 127

4Lesson Linking Equality and the Equal Sign Teacher Possible Learning Goals Look-Fors • Determines, through investigation, how to create ‘balance’ by creating sets of equal quantity • Explains or shows what ‘equal’ means, using concrete or pictorial representations, and starts to connect it to the equal sign • C reates balance on a pan balance in terms of quantity by removing or adding connecting cubes • C reates equations that match their actions • E xplains that balance is created by having the same number of cubes on each side • E xplains that the equal sign means ‘the same as’ or ‘balanced’ Math Vocabulary: A bout the ebpaeqaasquln,auamnabl,ctaaineolsaodnsnt,,,cenqmeqou,autattabnhclateh,iltiasnyna,gcmeed, Students have now investigated balance in terms of mass. For students to fully equation understand the meaning of the equal sign in number equations, they need to see equality in terms of quantity. Such experiences help to avoid misconceptions about the equal sign that can last and even grow in the later grades if not addressed. As Van de Walle points out, “after early introductions to the equal sign in grade one, the assumption is that students understand what you mean by ‘equals’” (Van de Walle & Lovin, 2006, p. 302). About the Lesson In this activity, students create balance between quantities using a pan balance and different coloured connecting cubes. They are also introduced to the equal sign. Materials: Minds On (15 minutes) pan balances, connecting cubes, • Review the previous investigations on balance. Ask students what created the chart paper, class anchor chart balance. Put three objects with the same size and mass on one side of the balance. Ask students what might balance them. Try out some of their Time: 45 minutes suggestions. Use the terms ‘balanced,’ ‘not balanced,’ ‘equal,’ ‘not equal,’ and ‘the same as’ throughout the discussions, clarifying their meaning. 128 Number and Financial Literacy

• Ask how many of the same objects would need to be put on the other side to create balance. (e.g., 3 objects) Ask why this would work, when in other cases two heavier objects could be balanced by five lighter objects. Discuss how the identical objects all have the same mass, so the same number of identical objects would create a balance. In these terms, it is the number of items that determines equality, since the masses of the individual items are the same. • Put 7 green connecting cubes on the left side of the balance and 5 red connecting cubes on the right side. Have students turn and talk with a partner about how they could make the two sides balanced or equal. (e.g., remove 2 green connecting cubes from the left side or add 2 red connecting cubes on the right side) Discuss their predictions and their reasoning. Ask students how they would describe the two sides now. Working On It (15 minutes) • H ave students work in pairs or small groups. One student can create an imbalance by putting a set of connecting cubes that are the same colour on one side of the pan balance or the other. The other student or students suggest how balance could be created using a combination of connecting cubes of two different colours. For example, one student may put 8 yellow connecting cubes on the right side and the other students put 5 green and 3 red cubes on the left side to create a balance. They can investigate four or five examples, taking turns to put the initial amount on the scale. Differentiation • For ELLs, when you use terms with the same meaning, such as ‘equal’ or ‘balanced,’ create a gesture with your hands each time so they can connect to a visual reference. • For students who need more of a challenge, have the first student place a number of cubes on the left pan and an unequal number of cubes on the right pan. Their partner then needs to create balance by adding or removing cubes in two different ways. Assessment Opportunities Observations: Pay attention to how students create equality. Do they pick up cubes and place them on one at a time to see what happens, or do they intentionally take a certain number of cubes all at once? In the latter case, they are predicting the change based on their understanding of relationships between the numbers. Conversations: For students working in a random manner, ask them to predict how many cubes they think are necessary and why. Scaffold the learning by asking questions such as, “Do you think it will be more than 5 or less than 5?” This helps students apply their number sense. Addition and Subtraction to 10 129

Consolidation (15 minutes) • Strategically select two or three groups to share one example each. Have them put cubes on one side and let the rest of the class predict what might go on the other side to create balance. After each example, ask students why the two sides are balanced or equal. (e.g., the number of cubes on each side is the same) Represent some of their examples in drawings. Put one of the drawings on the anchor chart that was started in Lesson 3. • Introduce the equal sign and explain that it means ‘the same as,’ ‘equal,’ or ‘balanced.’ Show how it can go in between the two sets of cubes drawn on the chart paper, reading it as, “7 cubes is the same as, or equal to, 5 cubes and 2 cubes.” Ask what makes them equal. (e.g., the number of cubes) Repeat this for the other examples and the example on the anchor chart. Highlight the equal sign and add a short definition. Further Practice • Reflecting in Math Journals: Have students draw one example of equality that they discovered using the pan balance and connecting cubes. Have them use words (e.g., balanced, equal) and symbols (=) to show the equality. 130 Number and Financial Literacy

5Lesson Representing Joining Problems Math Number Curriculum Expectations • B2.1 use the properties of addition and subtraction, and the relationship Teacher between addition and subtraction, to solve problems and check calculations Look-Fors • B2.4 use objects, diagrams, and equations to represent, describe, and solve Previous Experience with Concepts: Students situations involving addition and subtraction of whole numbers that add up have been exposed to the to no more than 50 plus and equal signs and know what they mean in Possible Learning Goals addition equations, and how an equation matches • Creates, represents, and solves ‘joining’ addition problems by acting a story. them out PMraotcheesmseast:ical rPerporbelseemntsinoglv,ing, • Explains or shows the equal sign as representing ‘the same as’ or a balance communicating • R epresents a given problem by acting it out and identifies the joining action Math Vocabulary: • C reates a joining problem and acts it out, making the joining action visible aeedqqduu,aaplt,ilouensq,,upmalluassticgshnigi,nnjgo, in, • S olves problem by counting on rather than counting three times equation • M akes an equation that matches the number story and explains what the numbers and signs represent • E xplains that the initial amount usually increases in a joining problem (unless 0 is added) About the Join and separate problems both involve an action that takes place over time. Join problems have the beginning quantity increased, while separate problems have the beginning quantity decreased. Marian Small points out that, “generally, but not always, students find joining and separating situations easiest to deal with” rather than part-part-whole and comparing problems (Small, 2009, p. 107). They are good starting points for addition and subtraction since the actions can be acted out or directly modelled, allowing students to visually see the change in the existing set. The context of the problem and how it is worded are therefore important. Fosnot and Dolk recommend that the contexts for addition and subtraction problems are closely connected to children’s lives and are designed to “anticipate and to develop children’s mathematical modeling of the real world” (Fosnot & Dolk, 2001a, p. 24). Stories can be created from recent common experiences, field trips, or discussions that occurred in other subject areas. Addition and Subtraction to 10 131

Materials: About the Lesson chart paper, markers, class In this lesson, students represent given problems through direct anchor chart modelling by acting them out. Students also have an opportunity to create their own addition stories using joining actions in a meaningful Time: 35 minutes context. In the Minds On problems, use your students’ names and change the situations so they relate to common experiences. Minds On (10 minutes) • Present students with the following problem. Modify the names and contexts so they reflect your students and their interests. − 4 children are playing on the swings. 2 more children join them. How many children are playing on the swings now? • Have students act it out. Ask how many children there are now. Explain that the ‘joining’ is like adding or putting together. Show them the addition sign and explain that it means ‘to join or put together.’ Gesture with your hands to make this clear. Have students act out the problem again and record a matching expression (4 + 2). Read it in relation to the problem (e.g., There are 4 children and 2 more come to join them. So 4 and 2 more are put together.). Ask how many children there are now. (You may or may not decide to add ‘= 6.’ The goal is to introduce the concept of adding or joining and linking it to the addition sign.) • Repeat this procedure with a second problem. Ask students how they could represent the problem with numbers and symbols, and explain what they mean in relation to the context. − 1 child is on the soccer field. 7 more children arrive to play soccer. How many children are on the soccer field now? • Have students show, with their hands, what they think the plus sign represents. Working On It (10 minutes) • Working in groups of four or five, have students create two or three of their own joining problems and act out the scenarios. Encourage them to be creative, using any materials in the classroom to help tell their stories. • Students may decide to try to represent the stories with numbers and symbols (on chart paper), although it is not necessary at this point in time. As you circulate, you may decide to scribe the number expression (e.g., 3 + 4) as they explain their problem to you, so they get familiar with the addition sign. • While students are working (or during the Consolidation), you could video students’ stories so they can be used later to review some of the adding concepts. 132 Number and Financial Literacy

Differentiation • For ELLs, they can act out a problem and the rest of the students can guess what is happening, offering the vocabulary to match the action. Assessment Opportunities Observations: Pay attention to students who cannot start the problem. Conversations: To scaffold the learning, ask students what they are supposed to do. To get them started, point to two students and ask what activity they could be doing. Have the two students do what classmates suggest. Then ask what might happen next if some students join them. Finally ask how the story ends. Consolidation (15 minutes) • Groups of students can act out their own problem. The rest of the class interprets what is happening and identifies what the joining action is. • After each group acts out their problem, ask the class what the matching expression (3 + 2) or equation (3 + 2 = 5) would be. Have them explain what it means in their own words. • Refer to the drawing added to the anchor chart in Lesson 4. Review what the drawing of the cubes represent. Ask students where they could put the addition sign to make their drawing complete. Write the complete equation below the drawing, annotating what each number and symbol represents. • Building Social-Emotional Learning Skills: Critical and Creative Thinking: Pose one of the following prompts to evoke some curiosity about adding and to connect it to their lives: – Ask students what they wonder about adding. (e.g., Can you join bigger numbers? Can you join 3 or 4 groups together?) – Ask what teachers or principals may want to add up in the school that might be very large. (e.g., how many students are in all the classrooms) Ask if they think it is possible. – Ask how they might add things in their lives. Reinforce the idea that when mathematicians are curious about math, they think creatively and find new ways of doing things. Addition and Subtraction to 10 133

6Lesson Investigating Strategies for Joining Problems Math Number Curriculum Expectations • B2.1 use the properties of addition and subtraction, and the relationship Teacher between addition and subtraction, to solve problems and check calculations Look-Fors • B2.2 recall and demonstrate addition facts for numbers up to 10, and related Previous Experience with Concepts: Students subtraction facts have created joining problems by acting them • B2.3 use mental math strategies, including estimation, to add and subtract out or using concrete materials. whole numbers that add up to no more than 20, and explain the strategies PMraotcheesmseast:ical used tcrPoeoropmolrsbemlaseuenmnnditscisnoatgrltvai,nitncegogg,n,iseneselecctitningg, • B2.4 use objects, diagrams, and equations to represent, describe, and solve situations involving addition and subtraction of whole numbers that add up to no more than 50 • B1.5 count to 50 by 1s, 2s, 5s, and 10s, using a variety of tools and strategies Possible Learning Goals • Solves addition (joining) problems by using a variety of modelling and counting strategies • Investigates the strategy of counting on • R epresents joining problems using concrete materials • S olves a joining problem in at least one way and explains or shows the strategy • C reates a joining problem that reflects the combining action About the Alex Lawson explains that students shift from acting out or drawing themselves in their solutions to representing the problem in ways that are less tied to the context, such as using cubes to represent people. While this is still direct modelling, Lawson stresses that it continues to be valuable since the concrete objects “decrease the amount of information that needs to be held in one’s head” (Lawson, 2016, p. 19). 134 Number and Financial Literacy

Math Vocabulary: When first solving joining problems, students often count three times. jroeipnr,egsreonutp,sv,issuuamli,ze For 5 + 3, for example, they first count out 5 objects, then 3 objects, push the sets together and count the combined set from 1 as their third count. Materials: With experience, students start to count on from one of the sets as they class anchor chart, include the second set (e.g., 5..., 6, 7, 8). This strategy becomes more connecting cubes, evident if the sets remain distinct after being put together (e.g., in two coloured tiles, two- different colours). This helps students reflect on the action after it has coloured counters, dice taken place so they “can see a relationship between the two parts and the (two per pair of whole” (Van de Walle & Lovin, 2006, p. 73). students), chart paper, coloured markers About the Lesson Time: 45 minutes In this lesson, students solve and create contexts for joining stories, but represent them using concrete materials of two different colours, giving them opportunities to develop different counting strategies such as counting on. Minds On (15 minutes) • Refer to the anchor chart started in Lesson 3 and review what the symbols and numbers represent in the equation under the drawing of the cubes. • Have students turn and talk with a partner and create a story involving students in the class that matches the equation represented by the cubes. Share two of the stories. Have students act out one of the scenarios. Ask how the cubes are similar to the people acting out the problem. (e.g., each cube represents one of the people) • Ask students how they could represent this problem using cubes: 4 red birds are in a tree. 2 yellow birds join them. How many birds are in the tree now? • Have a student model it using red and yellow cubes, while the other students explain how to represent it. Ask students how they might solve the problem. (e.g., count the red out, then count out the yellow, then count all) Ask if there is another way. • Ask how to show the problem in pictures and numbers, like on the anchor chart. On chart paper, draw cubes and record the equation, printing 4 and 2 in matching colours. (If students are still uncertain about creating matching equations, leave this part out so they can focus on solving the problem in several ways.) Working On It (15 minutes) • Students work in pairs. Tell them that they are going to create some problems using colours and numbers. Each student chooses a different colour and rolls a die. Together, they create a problem story using the two colours and two numbers (e.g., 5 blue fish are under a boat. 2 green fish join them. How many fish are under the boat?). Addition and Subtraction to 10 135

• Student pairs then represent and solve their problem with concrete materials of different colours. Encourage them to solve each problem in more than one way (e.g., While students are working, ask, “Is there another way to solve that problem?”). • If possible, students can record a matching expression (e.g., 5 + 2) or equation (e.g., 5 + 2 = 7) on chart paper. • Students create and solve one or two stories. Differentiation • For students who have not made the connection between the concrete and symbolic representations, have them just draw their problems on chart paper. As they explain the problem to you, record each number and connect it with the context (e.g., “So these cubes represent 4 blue birds, like this blue 4 here.”). Assessment Opportunities Observations: Pay attention to students’ counting strategies. Do they count all or count on? Do they refine or change their strategies according to the numbers, or use the same strategy every time? Conversations: • For students who count 3 times, counting out both sets and then counting the combined set starting at 1, use one of the problems that the student has modelled and explained to you (e.g., 4 red birds and 3 yellow birds join them): Teacher: So how many red cubes do you have in this group? Student: 4 (Does the student have to count or just ‘knows’?) Teacher: What are you doing with these yellow cubes? Student: Putting them with the red cubes. Teacher: So you have 4 red cubes…. (Pause and gesture toward the yellow cubes.) • If the student still counts the 4 red cubes one at a time: Teacher: You told me that there are 4 red cubes here, is that right? (Cover the red cubes with a piece of paper and gesture toward the yellow cubes.) • If the student still needs to count the hidden set of 4, he/she may need additional experiences with counting on and more time to be developmentally ready for the concept. Consolidation (15 minutes) • Strategically choose two or three pairs with solutions that reflect various counting strategies. Have each chosen pair tell one of their stories. Ask the rest of the class to visualize what their problem would look like. Pairs show how they represented their solution and explain how they solved it. 136 Number and Financial Literacy

• After each problem, ask if other students have another way to solve the problem. Have them demonstrate their strategy and why they chose it. Record each of the strategies that students used (e.g., counting all, counting on, counting on from the larger set), using pictures on an anchor chart entitled ‘Adding Strategies.’ • Together, create matching equations for the problems, colour coding the numbers so they match the cubes. Ask why we might use numbers and symbols to represent problems rather than words. Further Practice • Give students more opportunities to explore joining problems in a context using arithmetic racks. • Once students have had several experiences with joining problems using concrete materials, they can start to explore different mental strategies for addition, using visual representations. The visual images that students form in their minds can form the basis for developing and applying mental strategies when solving joining problems in numerical form. Materials: Math Talk: large and small arithmetic racks Math Focus: (or BLM 6: Blank Ten Frames and counters) • Representing joining story problems as part-part-whole relationships • Counting on Teaching Tip Let’s Talk Integrate the math talk moves (see Select the prompts that best meet the needs of your students. page 7) throughout Math Talks to maximize • V erbally present a series, or string, of word problems (adapt the context to student participation and active listening. the interests of your students). The first problem is expanded to model some of the dialogue that might unfold. • Pose the problem. There are 3 dogs in the yard. 2 more dogs come to play with them. How many dogs are there now? What kind of action do you hear in the story? (e.g., putting together, adding) Put your thumb up if you agree. Explain to your partner why you agree. With your partner, represent this problem on your arithmetic rack. • What did you do? (e.g., I slid across 3 beads on the top and 2 beads on the bottom.) Can somebody add on to what Zoe said? Put your thumb up if this is how you represented the problem. What do the 3 beads on top represent? (e.g., the first 3 dogs) Do I move them over in one move? Why? (e.g., Yes, because they are in the yard already.) Represent it on the large arithmetic rack. And what do the 2 beads on the bottom represent? (e.g., the 2 dogs that come in later) Do I slide the 2 beads across in one move? (e.g., No, because they come into the yard one at a time.) Does anyone have a different idea? Slide across the 2 beads. continued on next page Addition and Subtraction to 10 137

• Did anyone represent the story a different way? (e.g., I moved 3 beads across on the top and then slid over 2 more) What do your beads represent? (e.g., It’s the dogs, too, except I showed them altogether in one place.) Do both ways show the same action? Discuss their perspectives, resolving that both ways do represent the same story. • N ow what are we supposed to find out? (e.g., how many dogs are in the yard; I think there are 5 dogs now.) How did you get 5? (e.g., I counted.) How did you count? (e.g., I went 1, 2, 3, 4, 5.) Put your thumb up if you counted this way. Can you show us? Who has another way? (e.g., I know there are 3 on the top, so then I went, 4, 5.) Can someone explain in their own words what Thomas described? Why did you start at 3 rather than at 1? • Is there another way? (e.g., I just knew there were 5.) How did you know? (e.g., I saw 5 red beads together on the top, so I knew.) So you are saying that because the beads are in groups of 5, that the red beads on the top are 5. Is that right? (Yes.) • Let’s solve another problem so we can try out more strategies. You can make an anchor chart of strategies for one of the problems. Partner Investigation • C hallenge students to create more problems so they can find other ways to count. Have them take turns verbally posing a word problem to their partners, who solve it on the arithmetic rack. Together, they discuss whether there are other ways to represent or solve the problem. Follow-Up Talk • Meet after the Investigation to add any more strategies to the anchor chart. Materials: Math Talk: BLM 5: Ten Frames (0–20) using Math Focus: + 1, + 2 strategies for recalling addition facts to 10 quantities to 10 (or arithmetic Let’s Talk racks), counters, Select the prompts that best meet the needs of your students. class number • Select some of the following prompts to use in your Math Talk. You may line decide to cover different concepts (e.g., + 1, + 2) on different days. The following dialogue highlights both concepts to serve as an example. • Briefly show students different quantities up to 10 on a ten frame (or arithmetic rack). Below is a possible dialogue after showing 6 on the ten frame. • How many counters did you see and how did they look? (e.g., I saw 6. I saw 5 and 1 more on the bottom row.) Put your thumb up if you saw the same thing. • S how the ten frame again. How did you count the 6? (e.g., I counted 1, 2, 3, 4, 5, 6; I counted 5 and then 6.) Why did you start counting at 5? (e.g., I know there are 5 on the top, so I don’t have to count those.) Can anyone explain how Jamie knows? How could we show 5 and 1 more with numbers? (5 + 1) Where are 5 and 6 on the number line? (e.g., right beside each other, one number apart) Show us. 138 Number and Financial Literacy

• Visualize 1 more counter than 6 on the ten frame. How many would there be, and how do you imagine it? (e.g., I see 6 and then one more added is 7.) How did you know it was 7? (e.g., I counted 6…, 7) Put your thumb up if you counted on like that. What would that look like with numbers? (6 + 1) Where are 6 and 7 on the number line? • Did anyone see 1 more than 6 a different way? (e.g., I saw 5 and 2 more, so I counted 5…, 6, 7.) How could we represent this with numbers? (e.g., 5 + 2) Where are 5 and 7 on the number line? What do you notice? (e.g., they are 2 numbers apart) • Repeat with other numbers, connecting them to the counting sequence on the number line and to a number expression. Partner Investigation • S tudents take turns showing each other fast images of numbers on ten frames. Their partners say how many they saw, and what 1 more (or 2 more) would be. They can also say the matching expression (e.g., 6 plus 1 more equals 7). Follow-Up Talk • S how students some ten frames (or + 1, + 2 number expressions) and have them figure out the sums. What do you visualize? How can you prove your sum is correct? Addition and Subtraction to 10 139

7Lesson Addition: Commutative Property Math Number Curriculum Expectations • B2.1 use the properties of addition and subtraction, and the relationship Previous Experience between addition and subtraction, to solve problems and check calculations with Concepts: Students are familiar with addition • B2.2 recall and demonstrate addition facts for numbers up to 10, and related equations, the addition and equal symbols, and subtraction facts how they match a story. Students have worked • B2.3 use mental math strategies, including estimation, to add and subtract with arithmetic racks before (otherwise, they whole numbers that add up to no more than 20, and explain the strategies used can use connecting cubes). • B2.4 use objects, diagrams, and equations to represent, describe, and solve Teacher situations involving addition and subtraction of whole numbers that add up Look-Fors to no more than 50 • B1.2 compose and decompose whole numbers up to and including 50, using a variety of tools and strategies, in various contexts Algebra • C2.2 determine whether given pairs of addition and subtraction expressions are equivalent or not • C2.3 identify and use equivalent relationships for whole numbers up to 50, in various contexts Possible Learning Goals • Creates various contexts that represent addition (joining) problems • Explains or shows how the order in which the parts are joined in an adding problem does not affect the sum • C reates stories that represent ‘turning around’ addends • R ecognizes that the order in which two numbers are added does not affect the total • E xplains how ‘turn around’ stories such as 3 + 4 and 4 + 3 can take on different meanings in a story context, but still result in the same total 140 Number and Financial Literacy

PMraotcheesmseast:ical About the rPerporbelseemntsinoglv,ing, communicating Students may not understand the commutative property (that numbers can be added in any order, a + b = b + a) as they solve addition problems. For Math Vocabulary: join, example, in joining problems, the action takes place over time and two aemdqadut,acplh,luiensqg,upealqtuiuosanst,iigonn, similar problems (e.g., 2 birds join 3 birds, and 3 birds join 2 birds) can take on different meanings for young children. Carpenter and colleagues Materials: point out that, “children may not initially realize that two birds joining arithmetic racks or three birds gives the same result as three birds joining two birds” connecting cubes, (Carpenter et al., 1999, p. 8). Students need several experiences with dice (two per pair of creating and solving addition problems to understand that 2 + 3 students), large will always result in the same sum as 3 + 2. arithmetic rack or more connecting cubes, chart Minds On (15 minutes) paper • Students work in pairs with arithmetic racks. (Alternatively, if you do not Time: 40–45 minutes have arithmetic racks, have students work with connecting cubes of different colours.) • Tell students that they are going to create some joining story problems. Roll one die (e.g., roll a 2) and then roll the second die (e.g., roll a 3), so students can clearly see which number was rolled first. Ask for two or three stories that may match. Choose one to use as a focus (e.g., There are 2 birds. 3 birds join them. How many birds are there now?). • Have students represent the problem on the arithmetic rack (or with connecting cubes). Together, share how they solved the problem (e.g., counting all 3 times, counting on, ‘seeing’ a group of 5). Represent this problem on the large arithmetic rack by moving 2 beads over on the top in one move, and then 3 beads over on the bottom in one move (or with two different coloured sets of connecting cubes). • Ask how the problem would change if the 3 was rolled first rather than the 2 (e.g., There are 3 birds. 2 birds join them. How many birds are there now?). • Have students represent this ‘turn around’ problem with their concrete materials. • Show the first problem that you have represented and record the matching expression (2 + 3) on chart paper, explaining how the context matches the expression. Repeat this for the ‘turn around’ problem (3 + 2). Ask what the sum is for each. • Say, “It is interesting that they both equal the same sum. I wonder if this works with all numbers if we turn them around in an adding story.” Ask for thumbs up if students think so, thumbs down if they don’t think so, and thumbs to the side if they are unsure. • Tell students that they are going to further explore this idea. Explain that mathematicians think of possible rules (conjectures), but they have to find out if they work with all numbers to make them real rules. If they find one example that doesn’t work, then it cannot be a rule. Today, they will be mathematicians to see if the ‘turn around’ rule works with other numbers. Addition and Subtraction to 10 141

Working On It (10–15 minutes) • Students work in pairs. They each roll a die, create a story together, and then represent it on their arithmetic rack or with cubes of two colours. Next, they change the story so the second number is first, represent it with concrete materials, and solve it to see if they got the same total again. • Students repeat this for two or three examples. Differentiation • For students needing more of a challenge, have them choose their own numbers. They can also record matching expressions on chart paper. Assessment Opportunities Observations: Pay attention to any students who may find it challenging to make up or represent ‘turn around’ stories. They might find it easier working with cubes, since they have more freedom in manipulating them. Conversations: Ask students what the two numbers they rolled might represent. (e.g., dogs) Have them act out a scenario with concrete materials as you describe it (e.g., “3 dogs run into the yard [pause while they act it out with cubes] and 2 dogs run into the yard after them [pause]. How many dogs are in the yard?”). Now ask students to act out what would happen if the 3 dogs ran into the yard first. Have them describe the problem in their own words. Challenge them to try creating another problem with other numbers. Consolidation (15 minutes) • Have two or three pairs share one of their stories and explain whether they found the same solutions for their ‘turn around’ stories. • On two arithmetic racks or with cubes, show the two representations of one of their ‘turn around’ problems (e.g., 4 + 3 and 3 + 4). Ask how they are the same and how they are different. • Ask if anyone found an example that did not work. Ask if they think it will work for every number in the world, and whether they have proven the rule. • Building Social-Emotional Learning Skills: Critical and Creative Thinking: Discuss how students were real mathematicians as they tried many examples to prove a rule. Tell them that we think the rule works for now, but we won’t know for sure until we try it with even bigger numbers. Reinforce the message that they may not know if the rule always works YET, but with time, creativity, and critical thinking, they can learn more about it. Further Practice • Independent Problem Solving in Math Journals: Give students two matching expressions, such as 2 + 5 and 5 + 2. Have students draw a picture of a scenario that would match each expression. • In future related problem-solving situations, incidentally ask about whether the order in which the parts are added affects the total. 142 Number and Financial Literacy

8Lesson Addition: Varying the Unknown Math Number Curriculum Expectations • B2.1 use the properties of addition and subtraction, and the relationship between addition and subtraction, to solve problems and check calculations • B2.2 recall and demonstrate addition facts for numbers up to 10, and related subtraction facts • B2.3 use mental math strategies, including estimation, to add and subtract whole numbers that add up to no more than 20, and explain the strategies used • B2.4 use objects, diagrams, and equations to represent, describe, and solve situations involving addition and subtraction of whole numbers that add up to no more than 50 Algebra • C2.2 determine whether given pairs of addition and subtraction expressions are equivalent or not • C2.3 identify and use equivalent relationships for whole numbers up to 50, in various contexts Teacher Possible Learning Goal Look-Fors • Solves addition (joining) problems with contexts that vary the position of Previous Experience with Concepts: Students the unknown, using various tools and strategies have had experience with joining problems with the • C reates and solves a joining problem with either the change or start unknown two parts known and the • E xplains or shows how they solved the problem and how they know that their whole unknown. answer makes sense • E xplains the action that is taking place in the story and identifies what two parts are being joined together (or separated, if they worked backwards) • Explains their strategy using mathematical language (e.g., join, plus, equals) About the PMraotcheesmseast:ical Students are mostly exposed to joining problems that give the parts and the arPenrpodrbeplseroemvnitnsinoggl,v,inrega, soning accnoondmnsmetrcuatnitneicgga,ietsisenlge,cting tools whole is unknown. Problems with different structures, such as the change unknown (e.g., 6 + — = 8) or the start unknown (— + 3 = 7) are often overlooked. Marian Small states that students need exposure to a wide variety of these structures, so they develop “a more complete understanding of addition and subtraction.” She adds, “the different meanings for the operations can be distinguished only through appropriate contexts” (Small, 2009, p. 106). continued on next page Addition and Subtraction to 10 143

Math Vocabulary: The order in which the various types of problems are presented helps to aeedqqduu,aaplt,ilouensq,,upmalluassticgshnigi,nnjgo, in, scaffold the learning. Clements and Sarama highlight how the position of equation the unknown can affect the difficulty of the problems. For example, result unknown problems (5 + 4 = —) tend to be easier than change unknown problems (5 + — = 9), and start unknown problems tend to be the most difficult (— + 4 = 9), often using guess and check as the major strategy (Clements & Sarama, 2009, p. 62). About the Lesson In this lesson, students are exposed to joining problems in contexts that describe the unknown in various positions. The focus is on deriving meaning from the stories and clarifying understanding through discussion. Actually creating or understanding the related equations are not the goals of this lesson since this can take a long time to master. The equations are included to clarify the thinking of some students who find the numerical form easier to understand. You do not need to introduce the equations if your students are not ready for them. You may also decide to use the ‘story starter’ frames students are using in the Working On It instead. (See the sample ‘story starter’ below.) 5 and and 3 Now there are 7 Now there are 8 Materials: Minds On (20 minutes) chart paper, concrete materials • Pose the following story to students. (Change the problems by using your Time: 50 minutes students’ names and familiar situations, so students can connect to the context.) − “2 children are in the gym. 1 more child comes into the gym. How many children are in the gym now?” Ask how they would solve this story and what the matching equation would look like (2 + 1 = 3). Record the matching equation with a simple drawing that depicts the problem. Alternatively, you can record the problem using a ‘story starter.’ • Pose the next problem. − “3 children are in the gym. Some more children enter the gym. Now there are 5 children. How many children entered the gym?” Have them turn and talk with a partner about how this problem is different from the first problem (e.g., We know the final amount, but not how many are being added to make it.). 144 Number and Financial Literacy

Optional: If you decide to link it to a matching equation (3 + = 5) or a ‘story starter,’ record each part as you say, “3 children AND some more join them. Now there are 5 children.” Connect this to the previous question by covering up the 1 in 2 + 1 = 3. Ask them to put a context to it. (e.g., 2 children are in the gym. Some more join them. Now there are 3 children.) • Have students act out the problem to find the solution. • Pose the next problem. − “There are some dogs in the park. 4 more dogs come to the park. Now there are 6 dogs in the park. How many dogs were in the park at the beginning?” Have some students act it out. Share their strategies (e.g., Some students may work backwards thinking of 6 as the total and 4 as a subset, leaving 2 dogs at the start, or they may use trial and error.). Optional: Ask what the matching equation or ‘story starter’ would be (— + 4 = 6). Cover up the beginning parts of the two previous equations and have students put a context to them so they can connect the same meaning in all three scenarios. • Ask what is the same and different about all three types of problems. (e.g., Sometimes we know the final amount and sometimes we don’t. Sometimes we don’t know how many are at the beginning.) Working On It (15 minutes) • Students can work in pairs. Give students the following ‘story starters’ and have students create stories to complete them. You can print the story starters on chart paper. Students can represent the stories with concrete materials and solve them. • Have students create their own stories and record them using the ‘story starter’ format or pictures. (Students will be verbally sharing their stories, so they just need something to remind them of what they did.) 5 and and 3 Now there are 7 Now there are 8 Differentiation • You may want to only give ‘change unknown’ problems to students since ‘start unknown,’ as noted by Clements, are much more challenging to solve (Clements & Sarama, 2009). • For some students who are not ready for ‘change unknown’ or ‘start unknown’ problems, they can solve ‘result unknown’ problems. You can offer prompts to further the learning (see Assessment Opportunities). • For students who need more of a challenge, you can have them create the matching equations for their problems. Addition and Subtraction to 10 145

Assessment Opportunities Observations: Look for students who might just be extracting the two numbers and adding them together without paying attention to the context. Conversations: • Verbally give students a ‘result unknown’ problem to solve using concrete materials (e.g., 5 birds + 2 birds = ). Keep the two parts distinct (e.g., use differently coloured cubes). Teacher: How many birds are there now? How do you know? Student: 7. I pushed the 5 and 2 together and counted them up. Teacher: You say there are 7 birds. Imagine there are only 5 birds (cover up the two cubes). What has to happen so there are 7 birds? How do you know? Student: 2 more birds have to join them. They are under your hand. Teacher: So if I told you there are 5 birds and some more joined them and now there are 7 birds, how many birds joined them? Student: That would be 2. Teacher: (cover up the 5 cubes this time) What if there were 2 birds and more joined them, so there are now 7 birds? How many more birds came along? How do you know? • V erbally pose another ‘change unknown’ problem and have students figure it out with concrete materials. There is no need to attach an equation at this point. It is more important that students can use context to help develop new strategies. Teaching Tip Consolidation (15 minutes – 5 minutes for meeting with It may be beneficial to another pair and 10 minutes for meeting as a whole class) have the Consolidation the next day to break • Have students meet with another pair. They can take turns verbally sharing the up the lesson. story that they made up, while the other pair solves it. • Meet as a class. Selectively choose two or three stories that have the unknown in varying positions within the context. Focus on the different ways in which students solved the problems (e.g., Did they work backward? Did they guess and check?). Optional: As students share, record a matching equation. Ask students how each part of the equation connects to the story. • Ask what is the same about all of the stories. (e.g., They all involve joining two parts to make one whole.) • Building Social-Emotional Learning Skills: Positive Motivation and Perseverance: Ask students what problems they found hard to solve. Remind them of how much they have learned about adding in the past few days. Discuss what they learned from the mistakes they made and how mistakes can lead to trying new strategies. Reinforce the message that some learning is hard and they may not understand it YET, but by trying new strategies and not giving up, it will get easier. 146 Number and Financial Literacy

Further Practice • It is important to note that ‘change unknown’ and ‘start unknown’ problems offer much more of a challenge for many students and they will probably not understand them after one lesson. Try verbally presenting one simple word problem each day that varies the position of the unknown. For example, “Jen and Amit are at the round table. Some more students join them. Now there are 5 students. How many students came?” Students can act out the problem. As they act it out, ask them what the equation would look like and record it on chart paper. Keep adding equations as more problems are presented. On other days, identify one of the equations and have students make up a story that could match it. This activity can be completed outside of the math lesson when an appropriate context arises, or when there happens to be five free minutes. • Independent Problem Solving in Math Journals: From time to time, have students draw a picture of a given change or start unknown problem that you have posed to the class. Have them record the matching equation or you can help scribe it for them as they describe the action. Math Talk: Materials: Math Focus: Solving joining problems with the unknown in varying positions large arithmetic rack, small arithmetic Let’s Talk racks for partner use, chart paper, markers, Select the prompts that best meet the needs of your students. numbers 1–10 on small cards (one set per pair • P ose the following problem using an arithmetic rack, adjusting the context of students) to match the interests of your students: Teaching Tip − We want to have a collection of 8 books on 2 shelves. So far, we have 3 Integrate the math books on the top shelf. Slide across 3 beads on the top row. How many talk moves (see books will go on the bottom shelf? page 7) throughout Math Talks to maximize • Turn and talk to your partner about how you might solve this. Show your student participation and active listening. thinking on the arithmetic rack. Put your thumb up when you have a solution. • Possible strategies to discuss in greater detail: − Count 3 times: Recount the 3 beads, 1, 2, 3, then count on 4, 5, 6, 7, 8, while adding 5 beads to the bottom. Then counting the beads on the bottom, 1, 2, 3, 4, 5. − Same as above, except they ‘see’ the additional group of 5 (subitizing). − Comparing: Slide across 8 beads on the top row and 3 beads on the bottom row, visually match the beads on the top and bottom rows, and count the ‘leftovers’ as 1, 2, 3, 4, 5. • How did you solve your problem? How do you know your answer makes sense? Did anyone solve it in a different way? How are the strategies the same and how are they different? How could I finish off my representation to solve the problem? continued on next page Addition and Subtraction to 10 147

• R epeat with other numbers to represent total books and the original collection. • S lide 5 beads across on the top row and cover the entire top row (with a piece of folded cardboard) so students cannot see any of the beads. I have represented a certain number of books and hidden them under the cardboard. A student added 2 more books to the collection. Slide across 2 red beads on the bottom row. Now there are 7 books in the collection. How many books were there at the start? • Turn and talk to your partner and show your thinking on the arithmetic rack. Put up your thumb when you are done. • P ossible strategies: − Count on from the visible set by adding until they reach the total. − Build the total set on one row and the partial set on the other row. See the matching beads on the two rows and count the ‘leftovers’ in the total set. • H ow did you solve the problem? Can anyone show another way? How is your way like Jon’s way? Reveal the beads to confirm their thinking. Partner Investigation • Students work in pairs. Student A uses the top row while Student B uses the bottom row. Together, they choose a card from a deck of numbers 1–10. This will represent the total set. The first student slides across any number of beads that are less than the total on his/her row. The second student slides across the number of beads needed to complete the set on his/her row. Students confirm that the number of beads is correct by counting. • Students take turns being the first to slide the beads across. Follow-Up Talk • Discuss some of the strategies that students used. Record them on chart paper. 148 Number and Financial Literacy

9Lesson Representing Separating Problems Math Number Curriculum Expectations • B2.1 use the properties of addition and subtraction, and the relationship Previous Experience between addition and subtraction, to solve problems and check calculations with Concepts: Students have had experience with • B2.3 use mental math strategies, including estimation, to add and subtract addition and have been exposed to separating whole numbers that add up to no more than 20, and explain the strategies used problems. • B2.4 use objects, diagrams, and equations to represent, describe, and solve Teacher Look-Fors situations involving addition and subtraction of whole numbers that add up to no more than 50 PMraotcheesmseast:ical tcarPoeonropmodlrsbemplaseruonemnvdniitncssinaogtrgtl,avi,cnitnroegegng,a,niseseeoslcentciinntiggn,g • B1.2 compose and decompose whole numbers up to and including 50, using Msspeeuialgqqbutnuutshr,a,aatlpVtacislootkui,cngesman, asmb,iiwnjugoaualniatsny,cr,,,ehyamq:induigdna,ul,s a variety of tools and strategies, in various contexts equation Algebra • C2.2 determine whether given pairs of addition and subtraction expressions are equivalent or not • C2.3 identify and use equivalent relationships for whole numbers up to 50, in various contexts Possible Learning Goal • Represents ‘separating’ problems by acting them out or using concrete materials • C reates a story problem that involves the action of separation • S elects an appropriate strategy to solve a separating problem and explains or shows how it works • Identifies the whole and the parts in the story • M atches the parts of the equation to the story context • E xplains what the minus sign and equal sign mean About the Addition and subtraction are inverse operations. As Marian Small points out, since they “undo” each other, “any addition situation can also be viewed as a subtraction one, and vice versa” (Small, 2009, p. 107). Alex Lawson adds that while some curricula introduce subtraction well after addition, this does not benefit students. She stresses that “children who have a solid sense of cardinality and who have begun to develop an understanding of the part-whole relationship should be able to learn to subtract” (Lawson, 2016, p. 23). This is especially true when students use a ‘think addition’ or counting up strategy to solve problems, rather than a subtraction strategy. Addition and Subtraction to 10 149

Materials: Minds On (15 minutes) concrete materials (e.g., connecting • Propose the following problem, but change the names and the context so it cubes or counters), chart paper, markers connects to the experiences of your students: 7 children were playing in the park. 5 had to go home. How many children were left? Time: 45–50 minutes • Have students turn and talk with a partner and visualize the problem. Discuss their visualizations. Have some students act out the problem according to the instructions of the class. Ask how this problem is different from the other problems that they have acted out in the past. (e.g., People are leaving rather than joining the group; they end up with less people rather than more people.) • Introduce the minus sign as representing ‘removing or taking away’ as you create an equation that matches the scenario. Explain that the whole is being broken up into two parts. Have students identify the whole and the two parts in terms of the context. • Offer another problem to act out if students are still uncertain about the ‘removing’ action. Working On It (15 minutes) • Have students work in pairs. Have them create two or three story problems that involve the action of separating the whole. They can choose how they want to represent their problems (e.g., act it out, use concrete materials). Encourage students to be creative and to use materials from around the classroom to help tell their stories. Tell them that they can choose to create a matching equation for their story or wait until later to do so. Differentiation • ELLs may have difficulty transitioning from ‘joining’ to ‘separating’ problems presented in a context, not because of the math, but due to the language used in the stories. Offer a familiar but simplistic context using real objects (e.g., books placed on and off of a shelf) to enact an addition problem (e.g., 3 books on a shelf, 2 more added), and then enact a subtraction problem with the same materials (e.g., 3 books on a shelf, 2 taken away). Pair the language (e.g., adding, joining, subtracting, taking away) with the actions. Using the same context, describe a similar problem with different numbers and have them act it out. It is important to simplify the language, but not the math, since ELLs can capably learn with their peers when the language is no longer an obstacle to learning. Assessment Opportunities Observations: Observe whether there are students who continue to make joining problems and do not grasp the separating action. Conversations: If students created a joining problem, ask them to explain what is happening. (e.g., There are 3 students. 2 students sit with them and now there are 5 students.) Ask how many they are starting with in their story and whether they end up with more or less after the story is done. Ask how they could change their story so they end up with less at the end. (e.g., There are 3 students. 1 student leaves. How many students are left?) Reinforce how the action differs. 150 Number and Financial Literacy

Consolidation (15–20 minutes – 5–10 minutes for meeting with another pair and 10 minutes for whole-class discussion) • Have each pair of students meet with another pair. They take turns proposing their word problems to the other students, who then solve it by acting it out or using concrete materials. • Meet as a class. Since the possible learning goal is about representing separating problems, make connections between students’ representations by acting them out or using concrete materials. Select two or three pairs of students to present their problem to the whole class and explain how they represented it. Ask students how each of the problems can be represented using an equation. Record each equation as students give their explanation, pausing to show how each part relates to the context. Check for understanding by asking what each part represents. Save this list of equations for the next lesson. • Ask why we might use numbers and symbols to represent problems, rather than words. • Building Social-Emotional Learning Skills: Identification and Management of Emotions; Stress Management and Coping: Students are sometimes overwhelmed when they are first introduced to a new concept. They may get quiet and feel embarrassed if they make mistakes or feel confused. Ask how they are feeling, so they can identify their emotions. They can indicate the degree of their emotions using a thumbs-up, thumbs-down, or thumbs-sideways. Discuss how subtracting or ‘separating’ is a really new concept to understand and they have lots of time in grades one, two, three, and beyond to understand it. In the meantime, they can ask for help or try different strategies, even ones that don’t work. This will help them to develop resilience and learn not give up when stressed. • You can also raise students’ curiosity by asking what they wonder about subtracting stories. Encourage wondering since that is what makes mathematicians explore new ideas. Further Practice • Independent Problem Solving in Math Journals: Verbally pose one of the following prompts: – Draw a picture that shows addition. Draw a picture that shows subtraction. Write a matching equation for each. – Extending Understanding: Draw pictures to match 4 + 3 and 4 – 3. Show how they are the same and how they are different. Addition and Subtraction to 10 151

10Lesson Strategies for Solving Separating Problems Math Number Curriculum Expectations • B2.1 use the properties of addition and subtraction, and the relationship Previous Experience between addition and subtraction, to solve problems and check calculations with Concepts: Students have had experience with • B2.2 recall and demonstrate addition facts for numbers up to 10, and related addition and have been exposed to separating subtraction facts problems. • B2.3 use mental math strategies, including estimation, to add and subtract Teacher Look-Fors whole numbers that add up to no more than 20, and explain the strategies used Msspsmuialigbugatntnshtr,,castVjhaicogoitkinn,nceg,m,aeaeebiqwnqquuuuaulaasayal,,rtt,iyiamoeo:dnqnidn,u,uapsllus, • B2.4 use objects, diagrams, and equations to represent, describe, and solve Materials: situations involving addition and subtraction of whole numbers that add up concrete materials, to no more than 50 BLM 6: Blank Ten Frames, Algebra arithmetic racks, chart • C2.2 determine whether given pairs of addition and subtraction expressions paper, markers are equivalent or not Time: 40–45 minutes • C2.3 identify and use equivalent relationships for whole numbers up to 50, in various contexts Possible Learning Goal • Solves ‘separating’ problems using a variety of strategies and concrete materials • S elects an appropriate strategy for solving separating problems • E xplains or shows strategy using concrete materials • S olves the problem in more than one way • D escribes the action taking place in separating problems About the Lesson Now that students have had experience with creating and representing ‘separating’ problems, they can focus on investigating various strategies to solve the problems. Minds On (10 minutes) • Review the subtraction equations that were recorded in Lesson 9. Ask what action is happening in the problems. (e.g., taking away) Ask how the equation represents the various parts of the story. • Pose the following problem (change the numbers and context to suit your class): “There are 9 dogs. 4 run away. How many dogs are left?” 152 Number and Financial Literacy

• Tell students that they are going to solve this problem in as many different ways as they can. They can use various tools to help them solve the problem (e.g., ten frames, counters, arithmetic rack). Working On It (15 minutes) • Students work in pairs. Have them solve the problem in as many ways as possible, using concrete materials. They can draw a picture of one way they solved the problem so it can be shared with the class. Differentiation • If pairs of students can only find one way of solving the problem, arrange for them to meet with another pair that has a different idea. They can explain their strategy to each other and then continue working, either together, or back in their original pairs. Assessment Opportunities Observations: Pay attention to the various strategies that students use. Ensure that they are different strategies according to the thinking involved and not because they are represented with different concrete materials (e.g., cubes or counters). Students may: – Count three times: Count out a group of 9, remove 4 (counting 1, 2, 3, 4), and count how many are left (1, 2, 3, 4, 5). – Count out a group of 9, and then remove 4 as they count back 8, 7, 6, 5, keeping the 4 counters separate so they can see when they have removed enough (or keeping track of the 4 on their fingers). – Build 9 on a ten frame, remove the group of 4 all at once, and recognize the remaining counters as a group of 5. Similarly, students may slide across 9 beads on an arithmetic rack in one move (e.g., viewing 9 as 1 less than 10), seeing the group of 4 white beads, and then sliding them back, leaving 5 red beads. – Count out 4 counters, and then add more counters, counting on from 4 as 5, 6, 7, 8, 9, stopping when they reach the count of 9. – Count out 4 counters, add some more, and then recount to see if they have reached 9 yet. Conversations: At this early point, it is best to let students investigate on their own rather than imposing a strategy on them. Encourage them to find other ways or try using different materials or tools (e.g., “Would a ten frame help you solve this problem?”). They may choose to represent the same thinking with different materials, which is fine. Further input from other students in the Consolidation may help them broaden their thinking. Addition and Subtraction to 10 153

Materials: Consolidation (15–20 minutes) large and small arithmetic racks • Strategically select two to four solutions that reflect the various strategies (or BLM 6: Blank Ten Frames and counters) that students used (possibilities are outlined in Assessment Opportunities). • Have the creators of the chosen solutions show and explain one strategy (e.g., either their concrete models or their drawings). Ask the rest of the class if they solved it in the same way (they can show thumbs up), or whether they had another way of solving the problem. If students think they have another strategy, but they solved it in the same way using different concrete materials (e.g., representing it with cubes rather than counters), ask if the thinking is the same or different. Connect how the thinking is the same (e.g., they used counting three times, once with counters and once on the arithmetic rack). • As students share, create an anchor chart of their different strategies. Use drawings and annotate them. Alternatively, take photos of a solution that reflects each of the strategies and add them to the anchor chart. Further Practice • Show students the anchor charts for adding and separating problems. Discuss how addition and subtraction are different (e.g., In addition, we know the parts and find the whole; in subtraction, we know the whole and a part and find another part). Note: We cannot conclude that addition always ends up with a larger sum and subtraction always ends up with a smaller difference because this is not true for 0 or when working with integers. • Independent Problem Solving in Math Journals: Verbally pose the following prompt: − Draw pictures to show how 4 + 3 and 4 – 3 are different. Math Talk: Math Focus: • Representing separating problems as whole-part-part relationships • Exploring various strategies (e.g., counting back, counting up) Let’s Talk Select the prompts that best meet the needs of your students. • P ose some of the following prompts. • V isualize the following problem. There are 7 birds in a tree. 2 fly away. What do you visualize? What action is happening? (e.g., birds are going away) Do you think there will more or less birds in the tree? Why? (e.g., There will be less because birds are leaving.) • With your partner, show the problem on the arithmetic rack in at least one way. 154 Number and Financial Literacy