The session, Math Continuum: Addition and Subtraction - Session 3, was presented online on Tuesday, November 19th. This online session was presented by Heather Wark (Lakehead University), Alicia Schenck and Samantha Vanhooft (LKDSB).
Children should learn their number facts. However, they would benefit from learning these facts by using an increasingly sophisticated series of strategies rather than by jumping directly to memorization. - (Lawson, 2016, p. 4)
Strategies help students find an answer even if they forget what was memorized. Discussing math fact strategies focuses attention on number sense, operations, patterns, properties, and other critical number concepts. - (O’Connell & SanGiovanni, 2011, p. 5)
Strategies include: - Splitting - Getting to a decade # - Taking jumps of 10 forward/backward - Using overshoot & return
The session, Math Continuum: Addition and Subtraction - Session 1, was presented online on Tuesday, November 5th. This online session was presented by Heather Wark (Lakehead University), Samantha VanHooft, and Alicia Schenck (LKDSB Instructional Coaches).
Children should learn their number facts. However, they would benefit from learning these facts by using an increasingly sophisticated series of strategies rather than by jumping directly to memorization. - (Lawson, 2016, p. 4)
Strategies help students find an answer even if they forget what was memorized. Discussing math fact strategies focuses attention on number sense, operations, patterns, properties, and other critical number concepts. - (O’Connell & SanGiovanni, 2011, p. 5)
Strategies Include: - Counting 3x - Counting on/back - Counting on from larger # - Direct modelling vs. Tracking Join us for the next sessions that will dig into the strategies:
The session, Why We Use Lawson's Math Developmental Continua: An Overview, was presented online on Thursday, Oct 17th. This online session was presented by Heather Wark (Lakehead University), Sally Parkinson, and Kate Allison (LKDSB Instructional Coaches).
Children should learn their number facts. However, they would benefit from
learning these facts by using an increasingly sophisticated series of strategies
rather than by jumping directly to memorization.
- (Lawson, 2016, p. 4)
Strategies help students find an answer even if they forget what was
memorized. Discussing math fact strategies focuses attention on number
sense, operations, patterns, properties, and other critical number concepts.
- (O’Connell & SanGiovanni, 2011, p. 5)
Join us for the next sessions that will dig into the strategies:
Resources: Here are the links to the different folders for the games shared during the video that support students with Getting to a Decade Number and making jumps of 10 forward and backwards.
Math Game for Addition & Subtraction: Handful of Dice
Math Strategy: Up/Down over 10 Math Strand: Number Sense and Numeration
Overview: This game has many extensions and adaptations to suit your students' needs. With making pairs of 10, there are limited choices which let them focus on that amount. Then, by changing the number it allows students to practice decomposing other numbers of varying amounts. You can also change the amount of dice being used for each number. For example, you could choose the number 15 and have students use 2,3 or 4 dice to decompose it into. This is an excellent activity to support part-whole understanding. It also requires very little materials and could be played with 2-4 players.
* It could also be adapted for multiplication by giving a larger number, such as 24 and asking students to find pairs of numbers that when multiplied = 24.
How this activity supports learning: This game helps students understand that a number can be decomposed into smaller numbers. For example, to add 7 + 5, a child may use the up over 10 strategy and take 3 from the 5 and add it to the 7 to make 10, plus the remaining 2 = 12.
Composing and decomposing small sets of objects can allow young students to see the whole group as well as the subsets that created the whole
Students need a strong understanding of part-whole relationships in order to use mental mathematics strategies to add, subtract, multiply and divide.
Students who are able to fluently decompose quantities and then compose quantities are able to adapt their calculations to different sets of numbers.
Where to next? If a student is having difficulty decomposing, reduce the quantity of objects the student is using. Students may also benefit from having physical tools they can refer to, such as snap cubes or counters. For example, if the student is decomposing 5, have the, demonstrate the two parts with a 2 and 3 then take another 5 blocks and do it again with different numbers. You can ask if they are able to decompose it into other parts, such as 3 or 4.
If a student is fluent and confident with decomposing, move towards using larger numbers and having students explain their proof, “How do you know you have found all the ways to decompose this number?”
You can also focus on the next strategies, which would be splitting, where numbers are split into their place value to make them easier to add or subtract.
Strand: Number Sense and Numeration, Addition Math Strategy: Counting On
Video French:
Video English:
Materials: • 1 deck of cards with 10, J, Q and K removed • counters • 2 cups • 2 players
Overview:
In this twist on the classic War game, students work to find the difference in value between the two cards that are drawn. They are deepening their understanding of the part whole relationship and developing the counting on strategy, as they count on of from the lower card until they reach the value of the higher card. Students use counters to represent the difference.
How this Supports Student Learning:
Students are able to practice the counting on strategy in this game. Through this activity they construct their understanding of part whole relationships as they recognize the higher value card being the “whole”, the lower card being one “part” and the difference being another “part”. This game supports students in moving from counting all, or counting backwards into counting on.
Reflection & Consolidation
Which questions might you ask students to help connect their learning and consolidate their understanding?
Where to next?
For students that require more practice to master the counting on strategy activities like Salute, Riddles with Tiles and Dots and Number Dice provide excellent opportunities to strengthen this strategy. Once the student has mastered this strategy they are ready to begin playing games that promote the use of the 5 and 10 anchor, like Go Fish to Ten, Steal the Bundle and Make 10 Concentration.
Strand: Number Sense & Numeration Strategy: Up Over 10
Adapted from the updated e-text by Alex Lawson, ‘What to Look For’
Materials:
A deck of cards (Only use the 5, 6, 7, 8, & 9s)
Counters
Overview: The goal is to be the player with the most ‘change’ at the end of the game. With 2 or more players, split a deck of cards made up of the numbers 5-9. On each players turn, the player flips over 2 cards. They keep one number whole and decompose the other to make a ten with the number that was kept whole. The remaining amount is ‘change’. The change is the player’s points for that round. Students then take counters for that amount. This helps the students keep track of their score.
How This Supports Student Learning: This game will help solidify the Anchors of 10 and the Up Over 10 strategies. By decomposing one number to create a ten students’ will increase their flexibility with numbers and strengthen their understanding of the part-whole relationship between numbers. As well, the Up Over 10 strategy is a precursor to grasping ‘carrying’ that happens in the standard algorithm for addition.
Where to Next: When students are ready to extend this strategy, make the values of the cards drawn worth ten times the amount, as seen in the video. This will help them use the strategy with bigger numbers.
If your students are grasping the Up Over 10 strategy, encourage them to transfer the skill to Up Over other decade numbers. For example,
27 + 8
=27 + 3 + 5
= 30 + 5
= 35
Want a home extension? Send this link home with students and ask them to play Break Apart!
Strand: Number Sense and Numeration, Addition & Multiplication Math Strategy: Using Familiar Facts
Materials: • Triplets deck addition (BLM16 and BLM17) • Triplets deck multiplication (BLM19 and BLM20) (These are located in the Guides to Effective Instruction, Vol. 5) Overview:
Using the addition or multiplication cards provided in the GEIM Volume 5, students can play independently, in groups, or with the whole class. Students will have to match their answer card with the matching fact cards for each set. An example of a triplet would be the three cards: 3+5, 5+3, and 8; or 3 x 6, 6 x 3, and 18. In addition to sorting and matching the cards, you can also play a variation called « Go Fish », where students need 3 cards to make the set.
How this Supports Student Learning:
This game provides students an opportunity to strengthen their understanding of « commutative » property, which stems from the idea of moving stuff around. Addition and multiplication have the property of commutativity. When numbers are added or multiplied, the operation can occur with the values in any order and the same answer will be obtained: 3 + 2 = 5, 2 + 3 = 5; 4 x 6 = 24, 6 x 4 = 24. Subtraction and division are not commutative. Where to next?
Some students immediately see the connection between the commutative property of addition and multiplication. For others, this is something that has to be seen, felt and experienced many times before they believe it. That's okay! This activity provides an opportunity for the learners who need to see the connections to have that time. This property allows students to work with numbers flexibly, as they adjust numbers to make them easier to work with. One option to support students in grasping this concept would be to build arrays for multiplication and represent amounts with manipulatives when adding. Placing the values in the part-whole model would also support students to grasp this property.
Addition: Ten Frame activities in the Guide to Effective Instruction: Number Sense and Numeration K-3 support this thinking (i.e., Ten in the Nest, Ten Frame Cards, etc).
Multiplication: The game Boxed Out could support this thinking.
Fill the Tower - Use square tiles or linking cubes, take turns building towers on a game board (MK5). The towers are constructed by adding the number of tiles or cubes that have been rolled on the die. Play close attention to building the towers to the correct height. The game ends when one player successfully fills all the towers on his or her gameboard.
How this Supports Learning:
These games promote components of counting and the strategy of subitizing. There are three components of counting that children must learn in order to count accurately: one-to-one pointing (pointing at or touching each object that is being counted only once), the correct counting sequence (the sequence of number words you use to count 1,2,3,4,5,…), and coordinating (assigning a count to each object as you point at or touch it). As children learn to count accurately, they may have difficulty with any one, or all of these components. Counting accurately requires a great deal of time and many opportunities to count and to count for a purpose.
Overview of Games to support these strategies: What to Look For, pages 158-161
Where to next?
Once students have developed confidence with counting and subitizing they should move towards using the counting on strategy, which is when the student hold one of the numbers from an equation in their heads and continues to count on until they have counted as many as the second number. Games and activities to promote the Counting On Strategy can be found in the book What to Look For on page 162.
Fifty Chips- The object of the game is to be the first player to fill all 50 squares on the game board(MK4). Players take turns rolling the die, then puts that number of chips on their game board, one per square. The game ends when one player successfully fills all squares on his or her game board; or, when played cooperatively, when all game boards are filled. This game can be played as early as Kindergarten, but you may consider using game boards with only 20 or 30 squares, then giving each child 20 or 30 chips.
How this Supports Learning:
These games promote components of counting and the strategy of subitizing. There are three components of counting that children must learn in order to count accurately: one-to-one pointing (pointing at or touching each object that is being counted only once), the correct counting sequence (the sequence of number words you use to count 1,2,3,4,5,…), and coordinating (assigning a count to each object as you point at or touch it). As children learn to count accurately, they may have difficulty with any one, or all of these components. Counting accurately requires a great deal of time and many opportunities to count and to count for a purpose.
Overview of Games to support these strategies: What to Look For, pages 158-161
Where to next?
Once students have developed confidence with counting and subitizing they should move towards using the counting on strategy, which is when the student hold one of the numbers from an equation in their heads and continues to count on until they have counted as many as the second number. Games and activities to promote the Counting On Strategy can be found in the book What to Look For on page 162.
Tug of War- Using the Tug of War game board, roll the dice and "tug" the game piece to your side by moving it the number of places you rolled. Take turns "tugging" the game piece. The winner is declared when one player successfully moves the game piece to their side…winning the tug of war
How this Supports Learning:
These games promote components of counting and the strategy of subitizing. There are three components of counting that children must learn in order to count accurately: one-to-one pointing (pointing at or touching each object that is being counted only once), the correct counting sequence (the sequence of number words you use to count 1,2,3,4,5,…), and coordinating (assigning a count to each object as you point at or touch it). As children learn to count accurately, they may have difficulty with any one, or all of these components. Counting accurately requires a great deal of time and many opportunities to count and to count for a purpose.
Overview of Games to support these strategies: What to Look For, pages 158-161
Where to next?
Once students have developed confidence with counting and subitizing they should move towards using the counting on strategy, which is when the student hold one of the numbers from an equation in their heads and continues to count on until they have counted as many as the second number. Games and activities to promote the Counting On Strategy can be found in the book What to Look For on page 162.
Dot Bingo - In this whole class or small group game, students play bingo using cards that represent numbers 1-6 in a variety of ways (Numerals, dots, fingers, etc.). Games cards need to be constructed using the "What to Look For" Masters MK1 MK2. Game can easily be adapted for larger numbers, decimals. Let's play BINGO!
How this Supports Learning:
These games promote components of counting and the strategy of subitizing. There are three components of counting that children must learn in order to count accurately: one-to-one pointing (pointing at or touching each object that is being counted only once), the correct counting sequence (the sequence of number words you use to count 1,2,3,4,5,…), and coordinating (assigning a count to each object as you point at or touch it). As children learn to count accurately, they may have difficulty with any one, or all of these components. Counting accurately requires a great deal of time and many opportunities to count and to count for a purpose.
Overview of Games to support these strategies: What to Look For, pages 158-161
Where to next?
Once students have developed confidence with counting and subitizing they should move towards using the counting on strategy, which is when the student hold one of the numbers from an equation in their heads and continues to count on until they have counted as many as the second number. Games and activities to promote the Counting On Strategy can be found in the book What to Look For on page 162.
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!
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”.
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.
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 Strategy: Counting on from the Larger Number Math Strand: Number Sense and Numeration – Addition and Subtraction
Overview: In this strategy, students take Counting on to the next level. Counting on from the larger number means that you start with the biggest number in an equation, and then count on from there. For example, to add 5+3, students start with "5" in their heads, and then count up 3 more, "6, 7, 8." The counting on strategies works best when used for adding 1, 2, 3, or 4 to a larger number. If students try to count on with numbers higher than 4, it gets confusing, and mistakes happen because it is difficult to track. Counting on is an effective beginning addition strategy. Once students learn other more efficient strategies, they will begin using counting on in a more sophisticated way. For example, they will count on by jumps of ten instead of jumps of one. Or, counting on may be used with remaining addends after splitting and combining larger numbers.
How this supports student learning: When students count on from a larger number, they know that the larger number represents a whole set. Students no longer directly model the problem, but use their fingers as a way to keep track of a count. Being able to start at the larger number demonstrates the understanding of commutative property of addition where a pair of numbers can be switched around to reduce the amount of counting required, without changing the result, for example, 5+7=7+5.
Where to next? You can reinforce the Counting on from the Larger Number strategy by using activities that can be found in the book "What to Look For" by Alex Lawson, specifically on page 162. Once students are counting on from the larger number, support their development of the anchor of ten so they can combine their strategies of counting on and the anchor of ten.
Share your classroom experiences with Counting on from the larger number with us on Instagram and Twitter at @LKelempro #EngageLK!
Overview:
Taking from 10 is a strategy in the addition/subtraction continuum. It is further along the continuum because to use this particular strategy, students should be comfortable working with numbers.
This strategy involves students using ten-anchors to split numbers. They will take from 10, then add the leftover back on. For example:
Students have an understanding that 10 is a friendlier number, so they subtract from the 10, then add on the remaining from the start number back on.
Students are able to recognize they can subtract easily with a 10, or by making a 10.
How this supports student learning:
Once students have a strong understanding of using up/down over 10, they can begin to explore taking from 10 and splitting. As Lawson explains in the book “What to Look For” (page 44), this strategy helps to support splitting because students start to split double digit numbers along place-value lines (74 - 26 = 70 & 4, 20 & 6). Students can then subtract the decade numbers first: 70 - 20 = 50. They then subtract 6 from the result: 50 - 6 = 44. Finally, they add the 4 from the start numbers back on to the answer. This demonstrates how all of these strategies work together to support proficiency and student understanding.
** It is important to remember that this may be messy, but it is still a concept worth exploring with your class.
Where to next?
In the text “What to Look For”, the game Addition War on page 176 can be used to support a variety of addition and subtraction strategies.
There are also several mini-lessons in the Guides to Effective Instruction centred around supporting subtraction strategies
