Instant recall of times tables (multiplication number facts) are best taught using strategies for memorization. Ten Minutes a Day 2: book 2 teaches multiplication facts, and the Ten Minutes a Day 2: book 3 covers the division facts. Your child is ready to go. Ten Minutes a Day 2: book 4 covers both multiplication and division facts together revising all strategies taught – a great reinforcing tool, which increases speed and accuracy.
Good question, with no single right answer for every child. Here are some points to consider:
- Schools and teachers need to find out what their curriculum requires. In broad terms, the US and Australian curricula require all student to know up to 10x tables. The UK curriculum requires tables to be known to 12x. Teachers and schools will generally stick with the guideline minimum standards on that front.
- It is important to think about why a student would need to know beyond the 10x number facts. Looking through history it is easy to see why 12x number facts were essential: 12 inches in a foot, 12 pennies in a shilling. With metric measures (based on 10, 100, 1000, etc.) becoming the standard the world over, is it necessary any more?
- Some say 12x facts extend the mind. Fair enough, but why stop at 12x? Why not go for 13x, 14x, etc.?
- One line of argument is based on tradition: I learned it at school, so my child should do so too. You will have to decide if that is important to you.
- Dozens are a very common multiple and are still used, partly because 12 items can be arranged in 3 rows of 4, which saves on packaging compared to 2 row of 5, if we were to pack things in tens.
- Learning the 11x and 12x tables increases the number of facts to be committed to memory by 44%, compared to 1x-10x. Is it really worth that effort?
- 10x is essential for multiplication algorithms (sums) but 12x is not.
- 12x tables help with knowing hours in a day, and 360 degrees in a circle.
- If number facts are learned with strategies, students can work out the 12x answers (6×12 is the same as 6×10 + 6×2). Why the need to commit them to memory any more than 15x or 18x?
Because of different preferences and curriculum requirements, eBooks in our Number Fluency “Times Tables” Series which cover multiplication or division offer both versions, the 10x and the 12x. Make sure you choose the version you or your school prefer when you go to our store.
The use of ten frames help students with early addition and subtraction tables without the need for counting. To help students form numbers, use a blank ten frame and have students form the numbers with counters. Also use flash cards with the dots already printed on them to “flash” at them and have them call out the number shown.
These printed flash cards can be found in the Ten Frame Flash cards eBook with these patterns on them. Used daily with students instant recognition of the numbers without counting is established, initially to 10 then later 20. Use of the initial worksheet pages ( from the Ten Frames Worksheets eBook) helps establish the recognition and writing of all the numbers to 20. Playing the ten frame games (Ten Frames Games eBook) reinforce these numbers in a fun way.
Once students can instantly recognize numbers with no counting, teachers move onto the next step of using the 2 colored dot flash cards (found in the Ten Frames Flash Cards eBook). This card for 7 clearly shows that it is made up of 4 and 3. The related addition (3+4=7) and subtraction facts (7-3=4; 7-4=3) can also be established. Use can be extended to relating these cards to real life stories such as “4 boys and 3 girls, how many chidlren?” Have the students tell the stories about the cards. Use the worksheets to help guide you through this process. Identifying number fact families are essential for efficient memorization of addition and subtraction tables (number facts).
For more information on how to use ten frames watch our free 10 minute professional development clip here.
There are 2 recommended arrangements for ten frame counters: pairs and rows. There are advantages and disadvantages with both. Which one best used is up to the teacher or school. Parents are advised to go with the system your school uses. If students can cope with it, use both but don’t introduce them at the same time. Rather, teach one then, when established, introduce the other.
- The shape made with the counters appears to grow in size as the number gets larger. Compare 6 to 7 and it is visibly longer in length. This is less easily noticed with the rows arrangement as the length is already at its longest. It is only the bottom row that fills in. Many young children find this harder to identify.
- It is easier to see addition facts that make up the number. E.g. 7=4+3
- Odd and even are immediately identifiable.
- The numbers that make the pair to ten are seen more easily. 6+_=10 the missing 4 can be seen even without the counters there.
- Comparing numbers can be easier. compare 8 against 4 and the missing numbers (counters) can be seen quickly.
- The counters flow from left to right just like reading. When the top row is full then the next row is filled from left to right again.
- While the addition facts to 10 are not always as easy to identify, doubles to 20 are easier. As the numbers are larger than 5 and double 5 is 10; it is easy to see the 5+3 doubled is 16 E.g. 8+8but it is important to note the arrangement of the answer is in the pairs arrangement. The advantage to child being able to immediately identify both arrangements has a great over a child who can identify only one arrangement.
- comparing numbers is easy especially for numbers where both are greater than 5. It is less easy where one number is greater than 5 and the other less than 5.
To teach addition facts with ten frames it is important that the students already have instant recognition of the patterns of counters on the ten frame. For instance they must be able to recognize this as 6:
This is the “pairs arrangement” but the “rows arrangement” can be used also. It is even better if they can recognize both arrangements.
From this ten frame students can instantly see 4 + 3 = 7
But they can also see more. They can see that 3 + 4 = 7; 7 – 3 = 4; 7 – 4 = 3. They can also see 4 + __ = 7 and so on.
These are call fact families. Knowing fact families help students understand the relationship between one number and another. It reduces the number of facts or tables they need to memorize and help them with problem solving as the numbers are not remembered by rote but are remembered because the student can “see” it.
- Start by making these 2 color ten frames with counters and have your students tell you what they can see.
- Gently introduce mathematical language “four plus three” but only when the students are ready and it is age appropriate.
- Ask them to tell stories about what is shown on the ten frame. “There are 4 butterflies and 3more come and join them making 7 butterflies”
- Have the students show you a story you tell them using the ten 2 colored counters.
- Introduce take away stories.
- Change the stories slightly to increase complexity. “There are 4 cherries, how many more cherries are needed to make 7 cherries?” With practice students will begin to understand these are similar stories that have the same numbers in them.
- Once students are quick at using the ten frames and counters, introduce flash cards with the 2 colors on them. Flash the card and have your students tell or write (depending on age and ability of the students) the addition facts they can see.
- Flash the cards and have students write the fact families they can see.
Remember there should be no counting and no counting on fingers! It should all be done visually as students “know” what a number looks like and can recognize it instantly.
Understanding is the key. Introducing Fractions has dozens of activities, ones that teach understanding, not just the conventional coloring in activities so many book have. Introducing Fractions eBooks are designed for early years, but many of the activities and ideas can be used with older students as they present fractional concepts frequently not covered in many mathematics schemes. These concepts are crucial for fully understanding what is happening with fractions.
The 4 titles walk students through the important these concepts. Each title has an accompanying eBook of 30+ pages which has activities which can be worked through on screen:
- Fractions Counter
- Equivalent Fractions
- Fractions Converter
- Addition and Subtraction of Fractions
The on-line software allows students to enter most fractions and manipulate them whilst seeing what is actually happening through the process. Check out the free 30 day trial of the software and see for yourself how easy it is for students to grasp the concepts once understood.
Decimals are a specific form a fraction. Students must understand fractions before they can understand decimal fractions. Decimals are founded on our base ten system, so the fractions are specifically tenths, hundredths and thousandths etc.
Counting on fingers is a crutch that a child uses to get the right answer, BUT it is slow and inefficient. Children use this method because they can’t think of any other way to arrive at the answer without reaching for a calculator.
The problem with counting rather than having instant recall of number facts is that it takes the child’s mind away from the problem that they are working on and makes them think about how to solve the addition (or multiplication fact) that is needed, before they can go any further with the problem. In short it stops them thinking and complicates the question unnecessarily.
A child with instant recall of number facts is free to compare answers, think through a problem without having to stop and work out an answer in at a particular step, and is able to notice anomalies thinking, “That can’t be right.” In short, a child is armed with the tools to solve problems and work at a higher level of thinking, such as with fractions, decimals and algorithms, with ease.
Check out our number fact series that helps students become fluent in all the number facts they need. If you are after your child just knowing the basic 4 operations (addition, subtraction, multiplication and division) this Ten Minutes a Day 2 bundle is recommended.
A recent UK study “Testing times: which times tables do kids find the hardest?” showed that “The hardest multiplication was six times eight, which students got wrong 63% of the time. This was closely followed by 8×6, then 11×12, 12×8 and 8×12”. It is interesting to note that the 8x and the 12x feature prominently in this list. The grid below represents the relative difficulty of multiplication facts in a visual form. The blue are the correctly answered questions and the red the most incorrectly answered:
For example: 12x is just 10x + 2x the number. It is so easy to work out when 12x is thought of like that, but many students never make the connection. No-one has ever drawn their attention to this.
Another example: 8x is “double double double” – a little more tricky but not really that hard if practised regularly. The 6x facts are the 5x facts add one more set, so 6×7 is 5×7 + one more 7. Strategies give students a way of working out the answer when they have forgotten it. Rote leaves them with nothing but a guess.
The Bring It On! Fractions ebook is designed specifically for this task. This eBook gives daily practice with finding equivalent fractions, simplifying, adding and subtracting fractions, converting fractions form improper to mixed numbers, multiplying by whole numbers and converting to decimals. It is a great strengthener with knowing and using fractions.
Times tables are the multiplication number facts.
There are far more number facts that need to be known than just these.
- Addition facts to 20
- Subtraction facts to 20
- Multiplication facts (to x10 or x12 depending on the curriculum for your country)
- Division Facts
All of these are essential for fluency in number.
Most times table rhymes rote chanting systems do not teach why the answers are what they are, and offer no way of working it out if the number fact is forgotten. Using a system based approach such as that in the Ten Minutes a Day series, students are not only armed with instant recall, but also know why the answer is correct and a way of working it out if they forget. Not only that, but the strategies allow students to extend their knowledge well beyond the basic facts but use them with larger numbers, decimals, fractions and percentages.
The series is designed for use for just 10 minutes a day in a school classroom or at home. The Let’s Go! series have pictorial models and fewer examples on a pages reflecting the young age of the students using them.
The Ten Minutes a Day 1, 2 & 3 series teaches the core of the number facts with more questions, extensions and daily exercises. Revising number facts daily whilst extending them, guarantees students never forget. These facts aren’t just learned once then forgotten, but are refreshed daily, taking up only minutes out of the classroom timetable. The speed with which students can mentally calculate means classwork can be completed in a fraction of the time!
The Bring It On! series uses all the number facts that are fluently known from the earlier year levels and extends them into Fractions, Percentages, Order of Operations, and Mental Strategies for dealing with large numbers. Students are armed with skills for life.SCOPE AND SEQUENCE CHART
Learning your times tables is important. Keeping them fresh in a child’s memory is another.
Like 10 minutes of reading every day is essential for basic literacy, so is daily practice number facts essential for numeracy; yes 10 minutes every day, just like practicing scales on the piano before a lesson.
So often students are taught the basic facts in a few sessions, then they are left to themselves. Students need to be practicing number facts every day so that they never forget them even when older.
It is so common in older grade levels for teachers to have to teach number facts again rather than progressing on to higher level thinking such as fraction work or percentages. If students came to them with a fluency in number then their task is so much easier. And having students being fluent in all mental skills is the goal of all teachers pulling together as a team.
Fractions Converter Gadget is a great tool for helping students so the relationship between common fractions, decimals and percentages. It even shows that it does have a position on a number line, as well as the more difficult concept of ratios.
The eBook of worksheets that accompany it help you the teacher as there are lots of examples with all the hard work is done for you.
Calculators are a fantastic tool for everyone to arrive at answers especially when working with large numbers. It is easier and more reliable than using paper and pencil.
The biggest problem with calculators though is the user. Students are quick to type in numbers and then call out the answer. Few students ever stop to think about the answer given and whether it is correct or not. They never think to challenge the answer. Of course it not the calculator that makes the mistake, it is the user. What is typed in is critical to whether the answer is right or not. This is where knowing number facts come in. A student fluent with number facts, their extensions, fraction use, percentages and the like, can see immediately that the answer they have in front of them doesn’t make sense. It can’t possibly be right because answer is mathematical nonsense. There has been a mistake made somewhere and the student will have to go back over it and find the mistake that was made and recalculate the answer.
In short students who are fluent in their number facts is armed to think about whether the calculator answer is right or not.
My child is in gr 2 and struggling with understanding numbers. I want my child to know her addition and subtraction number facts fluently by the end of the year.
It is important stress not to move your child too quickly through the series unless they are coping well at each point. It is better to spend time focusing on the beginning stages of the ten frame work than rushing through with your child writing the correct answers in later books but having no idea what the numbers all mean. Once your child knows and understands early number the rest will fall into place.
Singing songs or rhymes with for each of the sets of times tables (or multiplication facts) is fun and enjoyable, but there are drawbacks to this method. When given a question such as 8×6, many students have to sing the whole song all the way through to get to the answer they need. This is slow and inefficient. Not only that, as students get older, going through a rhyme in their head can seem rather juvenile and therefore embarrassing. Another annoying feature is that if the student forgets the answer (gets stuck on the missing word) there is no way of working it out. Worse still is that fact that it slows down students’ thinking processes so that when in the middle of a complicated mathematical process, they have to stop, sing a rhyme then go back to whatever they were doing, if they can remember where they were up to.
The best method is to have instant recall of the times table that is required. Even better is to have not just that number fact on recall but all the related facts (e.g. 8×6, 6×8, 48÷6, 48÷8) known with them, instantly and accurately. Rhymes just don’t do that. Check out our Number Fluency Series which addresses this and will have all students armed with this skill.
Ratios are so important as they are used regularly in every day life. Making mistakes with ratio can cause incorrect medication or wrong concentrations of dangerous chemicals. Ratios matter!
But ratios are difficult because they are poorly understood. Most students do not realize that a ratio is another form of a fraction. The ratio however does not look like any of the other written fractions; for instance 1:3 does not look like 1/4 rather it looks like 1/3. Where does the the 4 come from?
Each of these ways of representing numbers shows different aspects of numbers, so it depends on what you are doing with the numbers as to which method is best.
What about paper and pencil algorithms: are they on their way out now that calculators and computers are here?
Algorithms, those pesky additions, long divisions and 3 digit x 3-digt multiplications we used to do over and over…
An algorithm is simply a standard series of steps to complete a repetitive task, which allows the task to be done quickly by just “plugging in” the numbers for each question.
Are they still relevant today? Well yes and no.
It is still important that students understand and are familiar with what is happening with numbers when we add, subtract, multiply or divide them. Calculators give students the idea that numbers are punched into a magic box and out comes an answer that is always right. They have no idea why or how it got there, or any idea of how it could be wrong. Essentially students expect calculators to do the thinking for them. This is a terrible mindset for students, never challenging or double checking that the calculator answer is right.
Students need to know what is happening to numbers and they need experience with efficient methods for working out a correct answer, but endless practice with pages of long calculations are no longer the goal in mathematics curricula. In the past this has been crucial in education as the only way to balance books and work out measurements etc was to be able to calculate accurately on paper. It was the only way to do it. Today this is no longer necessary and is where calculators and computers are a massive aid. But they are only an aid; the person entering the data has to know what they are doing and whether their answer makes sense. Trusting the calculator alone is a recipe for major errors.
We recommend that students learn strategies for thinking about their basic facts, as a way to quickly and thoughtfully learn number facts.
The good news is that all of these strategies will also help students to solve number problems. Many of them are based on understanding number and being able to make connections between them and the question at hand.
The Bring It On! Book 1 Mental Strategies has several introductory thinking methods which help students overcome common obstacles with thinking, such as ways to add a string of numbers, ways to double and halve large numbers quickly, multiplying by powers of 10, linking x25 to 1/4 of 100 etc. This idea of associating difficult ideas to harder ones can be learned and practised.
Other problem-solving strategies such as “reduce and calculate” (e.g., use simpler numbers than is used in the problem, solve it and see if the method is correct; and “guess and check” method) rely on the instant recall of basic number facts. The problem can be solved with smaller numbers quickly, processed and checked without the complication of going off and working out 9+8 or 4×7. Instead the process can be smoothly worked through efficiently and the real problem can be faced without necessary complicated outsourcing of steps.