Children are never too young to learn about treaties and their significance. In primary grades we can introduce students to the importance of agreements or promises, discuss how these agreements come about and the implications of breaking them. If we start the discussion by exploring what these ‘treaties’ or promises look like through their eyes, we can begin to deepen their understanding of their significance.
Use a simple text to provoke conversation among students in order to develop a broader understanding. For example, teachers could use the videos provided here as a starting point.
English:
French:
Teachers could also use mentor texts as the conversation starter.
English
French
A Promise is a Promise Robert Munsch
Turtles Race with Beaver Joseph Bruchac
Racoon’s Last Race Joseph Bruchac
Franklin’s Promise
Une promesse c’est une promesse Robert Munsch
After watching the video or reading a book, you could ask students questions such as:
English
French
What types of agreements or promises do you make?
Who did you make those agreements with or promises to?
How did the agreement come to be?
Can you think of a written agreement that you have been a part of? (i.e. class agreements)
What happens when that promise is kept?
What happens when you don’t keep that agreement or promise?
Quelles promessesfaites-vous?
À quilesfaites-vous?
Comment est-cequel’accorddevient?
Est-ce que tu peux penser d’un accord écrit duquel tu faisais partie?
Qu’arrive-t-il lorsque vous tenez votre promesse?
Qu’arrive-t-il lorsque vous ne la tenez pas?
Where to Next?
Create a promise together as a class. This could be a promise to treat everyone nicely at school or a promise to pick up litter and take care of the Earth. Whatever the agreement, discuss its importance with the students and how that agreement will be upheld.
Math Strategy: Anchor of 10
Math Strand: Numeration
Making 10 Concentration:
Overview:
This game is a variation of the concentration or memory game. This variation, from the original game, helps strengthen students’ understanding of the Anchor of 10 strategy. To begin place 16 cards face down between players. Players take turns flipping over two cards in an effort to find a match that adds to 10. If the two cards add to 10, the player collects the cards. If the cards don't add to 10, they are turned over again. When a pair of cards are removed, they are replaced with two new cards from the pile. At the end of the game, the player with the most cards wins. The game can also be adapted for the five anchor strategy, the same rules can be used to play Make 5 Concentration, using these cards: aces, 2s, 3s, and 4s.
How this activity supports learning:
Making 10 Concentration provides students with an opportunity to practice making 10. It is important for students to build a foundation of numbers using 10 as an anchor. It is a valuable strategy for computing basic facts and later larger numbers. With a solid anchor of 10 students will see that in 7 + 6 we can find a 10. They would take a 3 from the 6 to join it to the 7 to make 10. The remaining 3 is then joined with the 10 to make 13.
This strategy will later be carried over into multi-digit numbers such as 270 + 360. Students will see that in the 70 and 60 there is 100 with 30 left over. They would then be able to combine 500 + 100 + 30. Essentially, this is "making 10" which is an efficient way to solve computations, especially when doing mental math.
Where to next?
Laying the foundation of learning what makes 10 helps students move towards using up/down over 10 strategy. Look for more activities that support this strategy by looking in the book "What to Look For" page 174 - 184.
Share your classroom experiences with Making 10 Concentration with us on Instagram and Twitter at @LKelempro #EngageLK!
The main vision of the FSL curriculum is for students to “communicate and interact with growing confidence in French”. We know that in order for students to successfully learn a second language they need to use it. Second language teachers are always looking for new and engaging ways to get students communicating in the target language in order to purposefully practice the language in meaningful and purposeful situations.
Some of the best activities are ones that can not only be used to scaffold the language for students but then can be transferred to a variety of different learning contexts to provide these meaningful guided opportunities. Once students feel confident with their abilities and no longer need further practice, they can apply this newly acquired language to authentic situations.
Talking JENGA is one of those types of activities. It can be used as a way to give students a chance to practice necessary words and expressions, it can be used to create opportunities for students to ask and answer questions spontaneously in small groups, it can be used to as a prompt for conversation.
What do we need:
• Purchase one or several Jenga sets. Get them at garage sales, or pick up the smaller (more portable versions) at the dollar store.
• Number the blocks individually.
• Create a Talking JENGA Card to support the learning context of your choice. You can access a blank template here.
How to Play
The activity is best played in small groups to promote increased talk time during class. It is recommended 2-4 students per group. Students play JENGA as they regularly would, however, every time they pull a block, they must look at what number is drawn. From there, they will look to the provided Talking JENGA Card for the question they must ask their group members. Each player in the group must answer the question before the player puts the block on top of the tower. Then, the next player takes their turn and the process repeats.
This type of activity works great in small groups if you have enough JENGA sets for the whole class, or can be used for centres. It is a great activity to provide students at the beginning of class to get their Minds On in French. It can also be used as a challenge for those who finish their work early.
It is recommended to spend 5-10 minutes max on this type of activity, to maintain student interest and participation.
What do Talking JENGA Cards look like?
Practicing Necessary Vocabulary, Words or Expressions
The JENGA card could have each number linked to different images of the words and expressions that students are working towards internalizing. See the example here of colours and animals.
The numbers that could be linked to ‘picture sentences’. Essentially, images that you want the students to talk about in complete sentences with their group using focused language structures. For example, here students are learning to describe things. We are starting very basic with just colours and animals. The group could come up with the two sentences to the left. See the Describing Animals Set here.
If students are working on asking and answering questions more spontaneously, the numbers could be linked to a variety of questions related to the learning context. For example, if we are working towards students being able to talk about their families, a set like this one could be used.
A question set could be created based on an image that the group also has access to. For example, if they have a map in front of them like the one here and the question set could look like this.
The opportunities for this type of activity are endless. It is also great because the activities can be provided in a way that many entry points are possible. The activity can also be accommodated by providing students with a question set with sentence starters for prompts.
What other ways have you used this activity with your students? How would you adapt this to your students? Have you used this before? Tell us about how in the comments below.
Overview:
Playing games, like “Dot Plate Pattern Flashes”, is a great way to support subitizing to discover what strategies our students are using and where they are on the continuum of learning. The goal of this activity is for children to recognize the number of dots in front of them so that children can begin to identify the number by using familiar patterns, rather than by counting each dot one by one. By using a game to teach the concept, students will be engaged and this will allow the teacher to have a quick look into where each student is at with this strategy. If you’re looking for additional information around subitizing, check out our blog link below for the previous video “What is Subitizing?”
How this activity supports learning:
With students continually practicing their subitizing skills, they will start recognizing larger numbers by noticing small familiar patterns within the larger number. The more comfortable they become, the easier it will be for them to transition along the Student Continuum of Numeracy Development, from the “What to Look For” resource by Alex Lawson. There are also many variations that could support learning, such as asking students to tell you “one more” or “one less”.
Overview: Popular board games can be used to develop oral language skills with students. Game playing can be done with peers or teachers. It is also a great way to have fun and build relationships with both new students and more advanced learners.
Games can be purchased second-hand or brought in from home. Instructions for playing are the standardized ones that come with each game but can be modified to make the game quicker or easier to play. Cards could be chosen in advance based on targeted learning goals. New cards could also be found online, or developed for individual use.
How board games support students: Any early board games allow for peer tutoring and support oral language development and conversational skills. Some more advanced games also include some literacy.
Early Vocabulary building- Candyland for colours and numbers
Math Strategy: Various Addition and Subtraction Strategies
Math Strand: Number Sense and Numeration
Overview:
This game, located in the Guide to Effective Instruction (K-6, V.5, pg. 48), is used to support students development of multi-digit computations. As seen in the video, students need to determine how much is required to meet the target number. The game should be played whole class for a few rounds before students move to playing it independently. When learning the game, students should begin by playing with decade numbers. Once they are ready, move towards more challenging numbers.
How this activity supports student learning:
This activity would qualify as a "Where to Next" activity. If your students are developing an understanding of a subtraction or addition strategy they can use this game to practice. Most students require 3 to 5 interactions, with feedback, on a new skill/concept to develop an acceptable understanding. This game allows for students to practice a strategy while also receiving feedback from their peers. When it comes to a game, peers will be sure the feedback is given as it puts them in a better position to win!
Depending on the playing card numbers and the target number, this game could support the development of many addition and subtraction strategies seen on Lawson's Addition and Subtraction Continuum. It would work great with strategies like counting on/back, jumps forward or backward of 10/100, overshoot and return, constant difference, and getting to a decade number and taking jumps forward or backward. For subtraction situations, it would work well with splitting the subtrahend (2nd number in a subtraction sentence).
Students can use hundred charts, base ten blocks, and number lines to support their calculations. Remember, as students understand a strategy well they may begin to move away from concrete representations and move towards visual and numeric representations only. A calculator could be used to verify answers but not to find them.
Where to Next:
While students are playing the game observe the strategy they are using. If they are understanding the demonstrated strategy well then modify the game by giving them more challenging numbers. If appropriate, a student could be prompted to try the strategy without the assistance of concrete materials. Once students are using the strategy with grade appropriate numbers encourage them to learn a new strategy or combine it with a different known strategy for more efficient computations.
Good feedback contains information a student can use.
That means,
first, that the student has to be able to hear and understand it. A student can't hear something that's beyond his comprehension, nor can a student hear something if she's not listening or if she feels like it's useless to listen.
The most useful feedback focuses on the qualities of student work or the processes or strategies used to do the work.
Feedback that draws students' attention to their self-regulation strategies or their abilities as learners is potent if students hear it in a way that makes them realize they will get results by expending effort and attention.
Effective descriptive feedback focuses on the
ntended learning, identifies specific strengths, points to areas needing improvement,
suggests a route of action students can take to close the gap between where they are now and where they need to be,
takes into account the amount of corrective feedback the learner can act on at one time, and models the kind of thinking students will engage in when they self-assess.
These are a few examples of descriptive feedback:
You have interpreted the bars on this graph correctly, but you need to make sure the marks on the x and y-axes are placed at equal intervals.
The good stories we have been reading have a beginning, a middle, and an end. I see that your story has a beginning and a middle, just like those good stories do. Can you draw and write an ending?
You have described the similarities between _____ and _____ clearly, and you have identified key differences. Work on illustrating those differences with concrete examples from the text.
According to StatsCan, there are over 60 different Indigenous languages spoken across Canada. In our school board, there are 4 First Nations with 2 very different Indigenous Languages spoken, Lenape and Ojibwe. In this blog post, I am speaking about Ojibwe Word of the Week because that is what I have experience with, but these examples could definitely be used for Lenape Word of the Week.
Let me tell you a story: Last year a teacher at P.E. McGibbon (Nicole Gooding) said, “Hey! Let’s start doing an Ojibwe Word of the Week on our video announcements!” Nicole would find a new word each week, starting with greetings and then moving on to words based on the seasons and special events (eg. snow, rain, rabbit, egg). She, along with Chantima Olivera (Grade 1/2 ) would work together to get these words of the week spoken by children on their daily video announcements and onto the school’s Facebook page. Fast Forward a few months: Jen Gilpin from PE McGibbon and Chantima teamed up with Gretchen Sands-Gamble (that’s me) and Allie Kelly from A.A. Wright to form a TLLP (Teachers Learning and Leadership Project http://www.edu.gov.on.ca/eng/teacher/tllp.html). This TLLP team has created a blog to share our learning and understanding of implementing Indigenous Education and culture into their daily lives at A.A. Wright and P.E. McGibbon.
Where to Next?
If you’re wondering how you can include an Ojibwe (or Lenape) Word of the Week into your classroom or school there are many ways that this can take shape. Adding them to video announcements is what works at P.E. McGibbon. At A.A. Wright, a student goes on the announcements on Monday morning and says the Ojibwe Word of the Week for the whole school. A link to the Word of the Week Video is then sent out to each teacher in the school so they can share them in their classes.
Some classes create posters each week and put them up around the school so students are exposed to the Word of the Week when they are getting a drink or walking to the learning commons. Some people post the words to their Word Wall. Maybe instead of sharing through the announcements a class could take on the responsibility of sharing the Ojibwe Word of the Week through videos, emails and signs. The possibilities seem endless! What works at P.E. McGibbon or A.A. Wright might not work at your school, but keep trying until you find what works best for your school population.
Without staff members who are speakers of Ojibwe or Lenape, this can be difficult for most schools. A simple google search might not lead you to the word that you are looking for. There are many dialects of the Ojibwe language and the word you find online might not be the one that is spoken in your area. The best way to accurately find the words you want is to go to a local language speaker. If you’re having trouble finding a language speaker to help feel free to use the Ojibwe Word of the Week videos provided on the TLLP Blog: http://fne21c.blogspot.com/search/label/Ojibwe
Including the local Indigenous languages into your school day allows Indigenous students to hear and see their culture reflected and it gives them the space to feel belonging and meaning in their school. Non-Indigenous students get the chance to experience another culture’s language and share in the learning with the students and staff.
As always, if you need assistance send me an email! I’m here to help
It should be acknowledged that many of us are starting at varying entry points regarding our knowledge of the history of treaties in Canada and the treaty relationships between Indigenous People and the settlers of this country. I encourage all educators who teach in our schools to learn along with their students. This is an important aspect of Canada, and the treaties signed in what has become known as Canada have shaped this country and it would not exist as it does if it were not for treaties.
To deepen your own learning and understanding of treaties this is a magazine with many great articles about treaties, treaty relationship and what our next steps are as treaty people in Canada.
This resource is for teachers to use in the classroom. Your students can open this on their iPads to learn more about the history of treaties in Canada.
A great exercise is to relate the daily land acknowledgement (see an earlier blog post for the land acknowledgement video) to the treaties in the area. This website includes a map that shows you who traditionally lived on the land where you gather, work and live.
This link takes you to a PDF map that shows all of the treaties in Ontario by name and/or number as well as the First Nations in Ontario. Students can use this map to see what treaties are found in their area and then do research to learn more about that specific treaty. By living on treaty land in Canada, everyone is exercising their treaty rights. What treaties have benefitted you and your students? How have they benefitted your school population?
The Edugains website has some resources (lesson plans, links, videos) addressing First Nations, Metis and Inuit education. If you go to the following link, click on the "Resources" link. It will take you to lots of great resources.
A great picture book to use when learning about treaties is Hiawatha and the Peacemaker by Robbie Robertson. It is a story that is centuries old and tells of the Great Law of Peace and the Haudenosaunee.
Understanding the spirit and intent of treaties: When treaties were traditionally signed between sovereign Indigenous Nations there was a spirit of relationship between the two groups. When treaties were signed between the European settlers and the Indigenous people the two groups both had expectations that differed. The Indigenous People viewed the treaties as more of a covenant than a contract.
Contract/Covenant: What’s the difference? As a covenant, a treaty was seen as being based on good faith and good will, while a contract is meant to be negotiated and in search of the best deal. Contracts are based on written text and oral agreements while a covenant honours the spirit of the agreement. The Indigenous people viewed the treaties as a sacred commitment for both parties involved and recognized the spirit and intent as most important. This commitment was traditionally finalized by a sacred ceremony.
When reading about treaties, it is important to find accurate information on the intent of the original treaty as understood by the Indigenous community signing it. One of the sample questions from the SS/HG curriculum document is, “Why is it important to find accurate information on the intent of the original treaty as understood by the Indigenous community signing it? Why might there be differing interpretations of a treaty?”. It is important that teachers and students understand that there was a differing of views, and to keep in mind who’s voice is being represented in your research.
Treaty with Chippewas of Walpole Island 1857
"Fawn Island" Which was called, Keshebahahnelegoo Manesha
Expectations vs. Reality The expectations that the Indigenous People had regarding the treaties and what they actually received from the treaties differed very greatly. While they expected education, what they got was residential school. They expected health care in times of sickness, but they were given an Indian Agent who controlled the medicines and medical attention that was received. They expected to have their way of life protected but were given farming implements and hunting and fishing rights on reserve (All of which were again, heavily controlled by the Indian Agent). They had expectations of land sharing and in reality, were give small tracts of reserve land that were held in trust by the Crown.
It shows the treaties and the First Nations in our area. The First Nations are in the maroon areas.
When reading this map, consider the differences between land sharing and land ownership. Compare the words stewardship and dominion. How do these words and phrases differ? Have your students discuss these differences in relation to what the expectations of the Indigenous people when signing treaties and what actually received. What could or should have been done differently in the past? How are we all, as Canadians affected by these treaties?
Treaties: Not just a piece of paper In this video Janet MacBeth, Project Review Coordinator from the Bkejwanong Heritage Centre explains the importance of taking the spirit of the treaties into account. She reminds us that a treaty is more than a piece of paper, it is a recording of a living agreement that has many important aspects that go beyond signing the paper.
Where to next? Discuss agreements that you make in your classroom and the responsibilities that arise because of these agreements. Ask your students to think about what if these agreements are interpreted differently by varying people in the classroom? How can we relate this to the treaties that were signed by settlers and the Indigenous People? Write agreements with each other and ensure the spirit and intent of your treaties are recognized and respected.
As we are leading up to Treaty Week many teachers are looking for resources to support students learning. At this time we would like to acknowledge that many of us have learning to do around this topic, and we need to address this learning so we can move forward.
During this week we encourage students and teachers to learn, wonder and ask questions about treaties, treaty relationships and how treaties have shaped Canada as a nation.
“What is Treaty Week?” Treaty Recognition Week occurs in the first week in November and was first introduced in 2016. It was created to bring attention to the importance of Treaties and to help people from Ontario learn more about treaty rights and treaty relationships.
The Canadian Encyclopedia tells us :
"Indigenous treaties in Canada are constitutionally recognized agreements between the Crown and Indigenous peoples. Most of these agreements describe exchanges where Indigenous nations agree to share some of their interests in their ancestral lands in return for various payments and promises. On a deeper level, treaties are sometimes understood, particularly by Indigenous people, as sacred covenants between nations that establish a relationship between those for whom Canada is an ancient homeland and those whose family roots lie in other countries. Treaties, therefore, form the constitutional and moral basis of an alliance between Indigenous peoples and Canada."
From
You may have seen a Wampum Belt and connect that with treaties between settlers and Indigenous people. In Canada, it has been well documented that treaties took place between Nations hundreds of years before European contact. Many of these agreements were documented using wampum belts and are still alive today.
Wampum belts are one example of the way that Haudenosaunee, Anishinaabe and other Woodlands Nation’s history and culture is tied to oral tradition. For each wampum belt, there’s a certain set of promises that an assigned Carrier of the belt memorizes, and recites at the appropriate times. The design, and even the number of beads, acts as a mnemonic device, much like the picture-writing on the birch bark scrolls. The wampum keeper would hold the belt and read it by sight and touch and memory. The oral tradition of the wampum belts is a formal, highly allegorical diplomatic language. -Ojibwe Cultural Foundation 2011
This video explains the significance of the wampum belts.
This video explains the traditional meaning of treaties:
After watching this video this may be a prompt you pose this to your class:
Notice the language the elder uses,
“..It was to create a relationship where you worked in harmony and consulted with each other at what would work best for all nations, not just one or the other”.
What do you think this means?
Is this spirit of relationship and harmony reflected with the later treaties with the settlers?
Where to next?
The internet has many photos of Wampum Belts, and the Treaty Kit that each school was given in 2017 includes lots of information. Use this kit to help start your inquiry and research into what it means to be treaty people.
This infographic can be a great tool to start the inquiry process around treaties. What questions do you come up with as you read this infographic?
The revised Social Studies/History and Geography Curriculum document has many great sample questions to guide you and your students on learning more about treaties in Canada.
Stay tuned for more resources for your own learning and understanding of treaties and treaty relationships in Canada, as well as resources for your classroom!
Math Strategy: Trial & Error - Addition / Subtraction
Math Strand: Number sense and numeration
French Video:
English Video:
Overview
Trial and Error is a strategy that can be used when students are solving addition situations where the whole and one part are known, and one part is unknown. For example; Jack has some lollipops. Susie gives Jack 8 more lollipops. Now Jack has 14 lollipops. How many lollipops did Jack start with? The whole (14 lollipops) is known and one part (Susie gave Jack 8) is known. The student may use trial and error to find what the missing part is. The student adjusts their second guess based on what the outcome of their first guess is.
How this supports student learning
A student using trial and error to solve the lollipop problem will begin with the known part (8) and add another part to try to find the known total (14). The "guess" is not random but anchored to what they already know about number. For example, the student may reason that because they know that 8 + 8 = 16, the missing part could be 7. They will either use a concrete manipulative or counting on/all strategy to determine that 8 + 7 is 15 and is too high. They will then adjust their next guess based on what they discovered. The student is developing their understanding of part-whole relationships as they adjust the missing part to find the known whole.
Knowing the answer is too high, the student will adjust their answer and try 8 + 7
The student will again adjust their answer, based on what they discovered during the last trial.
Where to next
Whole group activities like "Mystery Number" are another way to practice the trial and error strategy.
An example of Mystery Number:
I have 7 candies and I get some more. Now I have 20. How more did I get?
Working as a whole group students can use familiar facts to help make reasonable predictions. For example, they make realize that 7 is 3 away from 10, so the mystery number is a group of 10 and a group of 3, or 13. Modelling this using concrete materials is a great way for students to visually see when they are over or under.
Share your classroom experiences with us onInstagram and Twitter at @LKelempro #EngageLK!
Math strategy: Using strategic and efficient strategies in addition, subtraction, multiplication & division
Math strand: Number Sense and Numeration
Overview:
Not all problems should be solved in the same way. Students need to have a variety of strategies that they can use proficiently when solving problems, yes, this may include the standard algorithm.
This handy checklist can help students know which strategy to use:
• easy to understand?
• easy to apply?
• easy to remember?
• easy to perform accurately?
With experience, students can learn to apply these criteria to their own strategies and develop ways for improving their efficiency and effectiveness.
Over-reliance on memorized addition and subtraction, multiplication and division procedures prevents students from using mathematical reasoning.
For example in subtraction students may persist with regrouping procedure to solve 2000-50, come up with 1050, and not reason that one step of the used procedure is missing, and their answer is wrong by a significant amount.
For example in multiplication a student using standard algorithm for 3x149 may come up with 327, forgetting to add the carrying numbers, and getting a wrong answer.
Students who learn the basic facts using a variety of strategies (i.e., making tens, using doubles, skip counting, using familiar facts) will be able to extend these strategies and their understanding of number to multi-digit computations and problem solving in more efficient ways. (Guide to effective instruction in mathematics, K-6, Vol. 5)
How this supports student learning:
Student-developed strategies are not necessarily efficient and effective.
For example, a student-generated strategy may require more steps and time than is appropriate. However, students develop and use these strategies because they make sense to them. As students share and compare their strategies, they will become better at finding methods that are both efficient and effective.
An efficient method is one that does not require a page of calculations and more than a reasonable amount of time to produce an answer. An effective method is one that works for all problems of a particular type (i.e., one that is generalizable to many problems using the same operation). Students need to learn to evaluate their own strategies and algorithms on the basis of the following criteria.
Is the strategy:
• easy to understand?
• easy to apply?
• easy to remember?
• easy to perform accurately?
Share your classroom experiences with us on Instagram and Twitter at @LKelempro #EngageLK!
Math Multiplication Strategy: Using Alternative or Standard Algorithms
Math Strand: Number Sense and Numeration
Overview:
The ability to use alternative or standard algorithms Indicates proficiency along the student continuum of Numeracy Development in Multiplication. Students benefit from working with a partial product algorithm before they are introduced to the standard multiplication algorithm.
Working with open arrays, as explained in the partial products video, helps students to understand how numbers can be decomposed in multiplication. The partial product algorithm provides an organizer in which students record partial products, and then add them to determine the final product. The algorithm helps students to think about place value and the position of numbers in their proper place-value columns.
When introducing the standard multiplication algorithm, it is helpful for students to connect it to the partial product algorithm. Students can match the numbers in the standard algorithm to the partial products.
The ability to perform computations efficiently depends on an understanding of various strategies, and on the ability to select appropriate strategies in different situations. When selecting a computational strategy, it is important to examine the numbers in the problem first, in order to determine ways in which the numbers can be computed easily. Students need opportunities to explore various strategies and to discuss how different strategies can be used appropriately in different situations. The standard algorithm is one of many strategies which students are eventually expected to perform.
Share your classroom experiences with us on Instagram and Twitter at @LKelempro #EngageLK!
Math Strategy: Doubling in Multiplication Math Strand: Number Sense Multiplication
Overview:
Students begin to use doubles in the early elementary years when learning addition facts. For example, students are doubling when they compute 3 + 3 = 6, 7 + 7 = 14, and 15 + 15 = 30. Soon they recognize that 3+3=6 is the same as 3 doubled (3 x 2)=6.
A situation where they may repeatedly add could be in a growing pattern.
As students begin to approach more involved math problems, they may accurately use repeated addition but the doubling strategy is more efficient.
Doubling can also occur when students see that a factor in a multiplication sentence can be broke into two equal parts:
The Guide to Effective Instruction in Mathematics Multiplication, Volume 3, page 23, has some information on doubling.
When students become aware of prime factorization they can use a more advanced form of doubling:
Note: The factors that work most effectively with doubling are 2, 4, 8, 16, 32...
How this supports student learning:
Through doubling, students are beginning to move away from addition and into multiplying. The nature of doubling promotes student thinking about the relationships between numbers.
WHERE TO NEXT?
In the younger years, promote the understanding of doubling when working with addition facts. Use terms like doubles, and two times the amount to connect to multiplication (Proportional Reasoning). Encourage students to use doubling in a ratio table to speed up the process when trying to find a term of a certain value. See if students can determine when doubling will not work in the different doubling situations noted above.
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Math Strategy: Using Partial Quotients Math Strand: Number Sense - Division
Overview: A Partial quotient is a partial answer to a division question. It is a step-by-step method of division wherein at each step, a partial answer is obtained. It allows students to work their way toward the quotient by using friendly multipliers such as tens, fives and twos without having to immediately find the largest quotient. After all the steps have been completed, the partial answers are added together to get the quotient. As students chose larger multipliers the strategies become more efficient.
How this supports student learning: The strategy of decomposing the dividend into parts (e.g., decomposing 128 into 100+ 28) and then dividing each part by the divisor is an application of the distributive property. According to the distributive property, division expressions, such as 128÷ 4, can be split into smaller parts, for example, (100÷ 4)+(28÷ 4). The sum of the partial quotients (25+7) provides the answer to the division expression.
The array is often used to model partial quotients. The dividend has been decomposed into numbers that are easier to work with. Consider the division expression 195 ÷ 15. Students can rework the problem into friendly numbers: 195 can be decomposed into 150 + 45, and each part can be divided by 15.
The same problem modeled in an open array to link the operations of multiplication and division. Students might decide to ‘multiply up’ to reach the dividend in order to find the quotient.
Students will need to use their factual knowledge in order to decide how to decompose the number. Students learn that facts involving 10 × and 100 × are helpful when using the distributive property. To solve 888 ÷ 24, for example, students might take a “stepped” approach to decomposing 888 into groups of 24.
WHERE TO NEXT? Students will use the flexible splitting of numbers and transition into flexible division algorithms, like the standard algorithm. With flexible algorithms, however, students use known multiplication facts to decompose the dividend into friendly “pieces”, and repeatedly subtract those parts from the whole until no multiples of the divisor are left. Students keep track of the pieces as they are “removed”.
Share your classroom experiences with us on Instagram and Twitter at @LKelempro #EngageLK!
What is Modeling Composite Units and Counting by Ones?:
Math Strategy: directly modelling groups of numbers and counting all by ones Math strand: Multiplication - Number Sense and Numeration
English:
French:
Overview: In the early stages of multiplicative thinking, students need to directly model problems in order to further develop their understanding. When using counting by ones, the student will count out the number of objects required in the group and will repeat this process until they have modeled the number of groups required for the problem. Finally, they count all of the objects in all of the groups from 1.
For example, if asked the product of 3 x 5, the student would count out 3 groups of 5.
Finally, they would count all of the objects by ones.
How this supports student learning: During the early stages of multiplicative learning, it is imperative for students to have multiple opportunities to develop their understanding of the operations. By using manipulatives to represent problems and physically sorting them into groups they are pushing their thinking from 'one to one' correspondence to a 'many to one' correspondence. This in turn will support students with their understanding of the big idea of unitizing (counting groups instead of individual items).
Where to next A next step is to model and encourage counting rhythmically so students can make the connection between the total number of objects represented in each composite unit.
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Math Strategy: Doubling and Halving Math Strand: Number sense and numeration
Location on the Continuum:
Overview: Doubling and Halving is a multiplication strategy where a student will create a simpler problem by doubling one factor and halving the other. For example, to solve 18 x 15 a student might:
Halving and doubling can also be represented using an array model. For example, 4x4 can be modelled using square tiles arranged in an array. Without changing the number of tiles, the tiles can be rearranged to form a 2x8 array.
The halving-and-doubling strategy is practical for many types of multiplication problems that students in the junior grades will experience. The associative property can be used to illustrate how the strategy works.
26×5=(13×2)×5
= 13×(2×5)
= 13 × 10
= 1300
In some cases, the halving-and-doubling process can be applied more than once to simplify a multiplication expression.
12×15=6×30
= 3×60
= 180
Where to next? When students are comfortable with halving and doubling, carefully planned activities will help them to generalize the strategy – that is, multiplying one number in the multiplication expression by a factor, and dividing the other number in the expression by the same factor, results in the same product as that for the original expression.
Source: A Guide to Effective Instruction in Mathematics, Kindergarten to Grade 6: Number Sense and Numeration, Grades 4 to 6 Volume 3 Multiplication
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Overview Trial and Error is a strategy that can be used when students are solving partitive division problems. In a partitive division situation students begin with the whole (for example 20 cookies) and the number of groups (for example 4 friends) and have to determine how many units would be in each group. Students who view this as a fair sharing problem will divvy up the total into each group, either one by one or in small quantities until the total is used up. When a student looks to use a multiplication strategy; making a prediction about the quantity in each group and then checking to see if they are correct, they are using the trial and error strategy. Students will often predict the number of items in each group is the same as the number of groups.
How this supports student learning A student using trial and error to solve the cookie problem will try out composites of different sizes. They may begin with 4 in each group, finding they have only accounted for 16 of their cookies. In trial and error, their guesses are systematic, not random. The next guess will be a direct reflection of the outcome of their first guess. Knowing 4 was too small they will select a new composite just slightly larger. Through this process, they are demonstrating and strengthening their ability to make reasonable predictions. The student is also reinforcing their ability to unitize, as they count each set of composite units (5 cookies) they understand that this represents one group (or unit).
The student begins by trying groups of 4. When they realize that they haven't accounted for all 20 cookies they will adjust their guess.
The student knew they were close, with 4 in each group, so they decide to try one higher. They count to find that 4 groups with 5 in each equals 20.
Where to next When playing card games, such as Go Fish to Ten begin with a set number of cards (for example 12) and ask students to predict how many cards each player will have. Activities like "Mystery Number" are another way to practice the trial and error strategy.
An example of Mystery Number:
I see 32 horse legs in a field. How many horses could there be?
The student recognizes that there are 4 legs on each horse
The student could begin with a known fact:
4 x 10 = 40.
Realizing this is too high they would go to another known fact, that is "nearby":
4 x 7 = 28
This time they are too low. They reason that the number of groups must fall between 7 and 10 and it must be 8, as they reason that 32 is closer to 28 than 40 and that 8 is closer to 7 than 9 would be. The student may also rely on a concrete or pictorial model to solve an unknown fact.
4 x 8 = 32
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