Math Strategy: Trial & Error - Addition / Subtraction

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.

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Math Strategy: Using strategic and efficient methods in both addition & multiplication

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)

http://www.eworkshop.on.ca/edu/resources/guides/Guide_Math_K_6_Volume_5.pdf 


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?

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Math Strategy: Using Alternative or Standard Algorithms in Multiplication

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.



How to Support Student Learning:
Multiplication Guide to Effective Instruction pages 20-21
http://www.eworkshop.on.ca/edu/resources/guides/NSN_vol_3_Multiplication.pdf

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.

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Math Strategy: Doubling in Multiplication

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.

http://eworkshop.on.ca/edu/resources/guides/nsn_vol_3_multiplication.pdf



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 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.



The Guide to Effective Instruction in Mathematics Volume 4 Division pages 18-30 has some very useful information.
http://www.eworkshop.on.ca/edu/resources/guides/NSN_vol_4_Division.pdf


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”.

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Math Strategy: Modeling Composite Units and Counting by Ones

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 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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Math Strategy: Trial & Error - Multiplication

Math Strategy: Trial & Error - Multiplication

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 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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Math Strategy: Counting on from the larger number

What is Counting on from the larger number?

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.

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Math Strategy: Taking from 10

Math strategy: Taking from 10 

Math strand: Number Sense

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

http://oame.on.ca/eduproject/ontariomathedresources/files/Number%20Sense%20and%20Numeration%20Vol%202%20Addition%20Subtraction%204-6.pdf

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Math Strategy: Using 5 X / 10X

Math strategy: Using 5 X / 10X
Math strand: Number Sense and Numeration - Multiplication and Division







Overview:
Students often learn to skip count by five and ten quite easily because of the wide variety of songs, poems, and storybooks based on fives and tens. It is important for students to connect the skip counting rhythm to concrete materials as they count. It is recommended to start with the ten times tables, then move to the five times table to help students see relationships between the products. For example, 5 is half of 10, so if you want to know what 5 x 6 is you could multiply 6 by 10 and then half the product to get 30.


How this supports student learning:
One way to think about multiplying by 10 is to think in terms of using place value units that are 10 times as big.


Another way to think about multiplying by 10 is to think in terms of a place value chart model. Where the numbers would shift one column to the left.




Notice that after multiplying a whole number by 10, there is always a 0 in the ones place of the product. This happens because after all the digits have moved left one place you end up with 0 ones. You can extend the same thinking to multiplying by 100 and 1000.

It may be tempting to simply give students rules about adding a zero to the end of the number when multiplying it by 10. However, it is important to focus on why adding zeros makes sense. With a firm foundation in why we add zeros, students will understand when it is appropriate to add zeros or when it isn't. For example, you don't add a 0 to the end when you multiply a decimal by 10 (4.2 x 10 = 42).

The 10 X strategy is an essential building block as many strategies for multidigit multiplication depend on decomposing numbers into their place value to multiply by 10, 100, or 1000. This strategy will support students’ understanding of multiplying decimals and measurement conversions. It will also support students’ understanding of dividing by 10.


Where to next?
Consider using The Big Race, an open worksheet, for targeted practice for 5X / 10X, as well as many other strategies, which can be found on page 69 in A Guide to Effective Instruction in Mathematics, Kindergarten to Grade 6 Volume 5

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Math Strategy: Repeated Subtraction

Math strategy - Repeated Subtraction
Math strand - Number Sense




Overview:
Repeated subtraction is the process of subtracting the same number from another number two or more times until 0 is reached. This strategy specifically supports division. To use this strategy, you take the dividend and repeatedly take the divisor away from it.



On the Lawson continuum, it is located in the Working With Numbers section.



Students may begin using repeated subtraction with manipulatives and then transfer this strategy to working with numbers. See video below:





With smaller numbers, this strategy is fairly efficient and effective. However, as students progress to larger numbers this can become challenging and inefficient.


How this supports student learning:
Repeated subtraction helps prepare students for the strategy partial quotients where they are taking off chunks of the dividend. When students are comfortable subtracting from larger numbers down to 0 with a constant amount, they will better understand how to subtract from larger numbers in different sized chunks. Below is an example of partial quotients to help you better understand.




Where to next?
Once students are comfortable with the repeated subtraction strategy, they should move towards practising partial quotients, which involves students solving a division problem by subtracting multiples until they get down to 0, or as close to 0 as possible. The subtraction of friendly multiples is key, such as using 5, 10, 25, 100 etc. Students will then add the multiples up to find the answer.

An activity that might help to support repeated subtraction would be simple subtraction activities from the Guides to Effective Instruction, such as any of the Learning Connections from the Counting unit, starting on page 143.

http://oame.on.ca/eduproject/ontariomathedresources/files/Number%20Sense%20and%20Numeration%201-3%20Revised.pdf


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Math Strategy: Using Familiar Facts

Math strategy: Using familiar facts 

Math strand: Number Sense and Numeration - Multiplication and Division









Overview:
Using this multiplication and division strategy, students use a known fact that is closely related to the unknown fact to help them solve the equation. For instance, the student who forgets the answer to 6 x 7 uses a known fact such as 5 x 7, and then adds on one more group of 7.




How this supports student learning:
This strategy builds on students’ prior knowledge to determine unknown facts. Students develop efficient strategies for fact retrieval when being encouraged to use reasoning skills and look for patterns and relationships between numbers. A student learns certain number facts before others, and they can use such known facts to derive answers for unknown facts. The most known facts are 1x, 2x, 3x, 5x, 10x and their doubles. So, if a student knows their 2x facts and they get a question like 4x6=? they can think… 2x6=12 so 4x6=24 since 4 is the double of 2.


Where to next?
Encourage students to look for familiar facts hidden within multi-digit multiplication questions and combine other strategies they have learned. See example below:




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Math Strategy: Empty Groups and Fair Sharing

How do we use the math strategy, 'Empty Groups and Fair Sharing' to divide?

Math Strategy: Empty Groups and Fair Sharing
Math Strand: Number Sense and Numeration

Overview:
At the beginning of using this strategy to divide, a student may share counters out equally by one into the empty groups. The number of empty groups is determined by the divisor. As they become more proficient students may share equally by other numbers (ex. 2, 5, 10). Students will determine the quotient when they count the number of counters in each group.

How this Supports Student Learning:

This strategy helps students see that division is about the whole (dividend) being shared equally. Creating the empty groups helps students identify the purpose of the divisor. Questions posed within a context allows students to connect their personal experiences to the mathematical structure of division. Students naturally understand fair sharing but do not naturally write this in a math sentence.

Where to Next:

As students grasp this strategy and begin to work with higher numbers, the teacher can encourage efficiency by challenging the student to use skip counting with larger intervals. For example, 150 divided by 3 could be shared out by 10s.



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Math Strategy: Ratio Tables

What is the math strategy, Ratio Tables, all about?

Math Strategy: Ratio Tables
Math Strand: Number sense and numeration



Location on the Continuum:




Overview:
Ratio tables are a way for students to model their thinking. The structure of a ratio table allows students to find and use multiplicative relationships and equivalent ratios to solve problems in various ways.


How this supports student learning:
Ratio tables promote mental math strategies in a way that resonates intuitively with students, given the structure of the ratio table itself. Some students develop comfort and sophistication with doubling and halving strategies. Some students rely heavily on multiplication by 10’s. Some students prefer additive strategies rather than multiplicative ones. The point is that the ratio table fosters mental math strategies, but in a context and through a structure that supports the child’s development of mathematical understanding.


In the example of 18x5, the student could:

Begin with a known fact, 1x5


Use doubles to solve 2x5


Double 2x5 to find 4x5


Combine 1x5 + 4x5 to find 5x5


Double 5x5 to find 10x5


Combine (1x5 + 2x5 + 5x5 + 10x5 = 18x5) to find 18x5



Where to Next?
Once student are comfortable using a ratio tables, encourage them to continuing exploring doubling and halving and partial product and quotient strategies to move towards proficiently in multiplication. 

Links to support the use of Ratio Tables:

https://www.mathisfigureoutable.com/proportional-reasoning-with-ratio-tables-1/

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