Homeschool Math Blog

Someone asked for help to explain the concepts of multiplicative word problems involving "as many as", like the ones below: 1) Haley had four times as many dollars as her sister. Together they had $60. How much money does Haley have? 2) Rachel had 5 times as many dollars as her sister, Nora. They had a total of $90. How much money did each of them have? The BAR MODEL is an excellent tool for helping children understand what is going on in these types of word problems. In (1), draw a bar for Haley and another for her sister. Divide Haley's bar into four parts, and make the other bar just one such part long. Haley |---|---|---|---| Sister |---| Now you will see that the TOTAL needs divided into FIVE equal parts — and from then on it is easy-peasy. Additionally, you can use Thinking Blocks website to build such bar models INTERACTIVELY. For problems like (1) and (2) above, choose the fourth model from the left (rightmost) in the top row that is b...

I have been working in a particular section of my HomeschoolMath.net site - the section that has free math lessons. It's been needing some updating for a long time... and finally I got around to it. Websites are like most anything else you own - you have to take care of them. Well, they CAN go on their own (like autopilot) for a while but if you never update them, they will eventually lose traffic and go downhill. I still have lots of work to do... (and I'll do my best to keep cleaning and improving the pages in this section) but here are some updated multiplication and division lessons. They are essentially sample lessons from my books. Enjoy! Multiplication Lessons Grade 3 Multiplication concept as repeated addition Multiplication on a number line Multiplication is commutative Multiply by zero Multiplication word problems Order of operations Grade 4 Multiplying by whole tens & hundreds Distributive property Partial products - the easy way Multiplication...

The two videos below show how you could teach multi-digit multiplication, or the multiplication algorithm, or multiplying in columns to students. Teaching multiplication algorithm Multiplication algorithm with a 2-digit multiplier I approach this in steps. First, to teach students to multiply 4 × 87 or 5 × 928 (one factor is single-digit): 1) Teach students to multiply single-digit numbers by whole tens and hundreds. 2) Teach them the partial products algorithm; 3) Use the above as a stepping stone and teach the usual multiplication algorithm. Then we can go on to the two-digit multiplier: 4) Teach the partial products again. 5) Teach the regular form of the algorithm. Let's look at these steps in more detail. Step 1. This means teaching students to multiply 5 × 80 or 7 × 400 or 3 × 40 or 9 × 900 (mentally!). The shortcut is to multiply without the zero or zeros, then tag the zero or zeros to the result. But, where does it come from? For example, 5 × 80 is the same as 5 × 8 × 10. ...

In this video I show, first of all, the common shortcut: you move the decimal point in the number as many steps as there are zeros in the number 10, 100, 1000 etc. For example: 2.16 × 10,000 = 21,600.0 It is as if the point moved four steps from between 2 and 1 to between zeros. You can see better examples of this in my lesson Multiply and Divide Decimals by 10, 100, and 100 at HomeschoolMath.net. Then, I also show where this shortcut originates , using PLACE VALUE charts. In reality, it's not the decimal point moving (it's sort of an illusion), but the digits of the number move within the place value chart (to the opposite direction from the way the decimal point seems to "move"). This explanation can really help students to understand the reason behind the "trick" of moving the decimal point. Multiply & Divide Decimals by powers of ten

I will be hosting the blog carnival Math Teachers at Play next week. (You can send in submissions here .) One submission I got about various multiplication tricks or shortcuts got me inspired to write a proof of the particular trick. You could definitely use this in algebra class. First explain the shortcut or trick itself. Then ask students to prove it, or to explain WHY it works, using algebra. You could also explain this to younger students as an additional "neat trick" and let them explore and play with it. THE "TRICK" If a number ends in 5, then its square can be calculated using this "trick" (I like to call it a shortcut because there's nothing magic about it): Let's say we have 75 × 75 => Go 7 × 8 = 56. Then tag 25 (or 5 × 5) into that. You get 5625. Let's say we have 35 × 35 => Go 3 × 4 = 12. Then tag 25 into that. You get 1225. Let's say we have 115 × 115 => Go 11 × 12 = 132. Then tag 25 into that. You get 13,225. Let...

This is a tough topic... in a sense. It is not difficult at all, if you just follow the rule given in your math textbook, because the rule is pretty straightforward: To multiply decimal numbers, multiply them as if there were no decimal points, and then put as many decimal digits in the answer as there are total in the factors. The difficulty is only if you try to understand why we have such a rule - where does it come from? Understanding the rule for decimal multiplication is actually fairly simple, because it comes from fraction multiplication. But, I will propose here a little different way of explaining all this. First, look over this decimal multiplication lesson that is taken from Math Mammoth Decimals 2 book. It talks about how 0.4 × 45 is like taking 4/10 part of 45. The same applies if you have 0.4 × 0.9 - you can think of it as taking 4/10 part of 0.9. Can you see now why the answer to 0.4 × 0.9 has to be smaller than 0.9? Or, turn it around: 0.9 × 0.4 is taking 9/10 of 0....

The information below is from another Maria, namely MariaD from NaturalMath.com. I'm posting it here with her permission, as it might interest some of my readers. Hello! My name is MariaD, and I love multiplication. Natural Math is starting a research and development family group about this topic. You are cordially invited! Please forward this invitation to other families who may want to join. There are three main benefits. You receive individual family math coaching. You access a community of other parents sharing questions and ideas. And you contribute to a beautiful and much needed web resource for the future. There are two main responsibilities. At least weekly, you will run custom family math activities you select. As needed, you will talk with me or other group members about your activities. We can talk by email, chat, voice, or face-to-face in Cary, North Carolina, USA. At this early stage, we need active group members. If you plan to be a quiet fly on the wall, please wait ...

Keith Devlin has published another column along the lines of multiplication not being repeated addition . I feel quite honored that he mentions THIS blog in his column (scroll down to the end), referring to what I wrote about the issue. This time he expounds on research results. The research clearly shows that thinking of multiplication as repeated addition hinders students' further understanding of mathematics. It can lead to the misconception that multiplication always makes things bigger. Children need to acquire multiplicative reasoning, which is different from additive reasoning. And so on. Go read it yourself.

It's been very good and educational for me to refine my thinking on multiplication vs. addition by reading some recent posts around the blogosphere, especially What's wrong with repeated addition by Denise and Devlin's Right Angle Finale at Text Savvy. I feel that on some blogs people aren't even exactly talking about the same thing. The subjet we're dealing with - is multiplication repeated addition or not? - is subtle. Some people talk about how to define it - it is defined in some systems as repeated addition, and they feel that closes the issue. BUT, I tend to agree with what Denise wrote: multiplication is a different operation from addition and somehow we need to get students to view it that way. I've always known that; I've never thought anything different. But yet how we present things to children is not always easy; we may understand the idea but not able to convey it right. Talking about multiplication as repeated addition MAY indeed leav...

I just found out an interesting column by Keith Devlin ... he tells elementary teachers to stop telling the students that multiplication is repeated addition. Why? His point is, this idea does not carry through. As soon as the student encounters multiplication of fractions (or of decimals), it won't work. You can't think of 3/4 x 6/11 as repeated addition. He feels it's better to portray multiplication as a scaling process: say 5 x 9 means 9 is scaled by a factor of 5. Then, students can have a true "aha" moment as they discover for themselves that you CAN use addition to find the answer to 5 x 9. But, Devlin says, they should be taught and shown the multiplication idea as a scaling process. Now, I feel that Devlin has a point here... so since I'm constantly in the process of writing math materials for my Math Mammoth series of books, and right now I'm writing lessons on multiplying decimals for 5th grade, I took this idea just yesterday and tried to go wi...

Some recent additions to Math Mammoth Blue Series books: Math mammoth Multiplication 2 This book concentrates on multi-digit multiplication, first explaining what it is based on (multiplying in parts), then practicing the algorithm. also included: order of operations, multiplying with money, and lots of word problems. See sample pages here: (PDF) Contents & Introduction Multiply by Whole Tens and Hundreds Multiply in Parts Multiplying in Columns, Standard Way Error of Estimation Order of Operations Money and Change Multiplying 3-digit by 2-digit Math Mammoth Division 2 This book includes lessons on division, long division, the remainder, part problems, average, and problem solving. See samples: Contents and Introduction Division Terms, Zero and One Finding Parts with Division Long Division 1 Long Division with 4-Digit Numbers Average Divisibility Rules NOTE: Multiplication 2 and Division 2 now replace the earlier book called Multiplication Division 2. Math Mammoth Place Val...

I haven't blogged for a while but I've been thinking about this topic for a little while now. It is your multiplication algorithm, also called long multiplication, or multiplying in columns. I also happen to be writing a lesson about it for my upcoming LightBlue series 4th grade book . The standard multiplication algorithm is not awfully difficult to learn. Yet, some books advocate using so-called lattice multiplication instead. I assume it is because the standard method is perceived as being more difficult. But let's look at it in detail. Before teaching the standard algorithm, consider explaining to the students multiplying in parts , a.k.a. partial products algorithm in detail: To multiply 7 × 84, break 84 into 80 and 4 (its tens and ones). Then multiply those parts separately, and lastly add. So we calculate the partial products first: 7 × 80 = 560 and 7 × 4 = 28. Then we add them: 560 + 28 = 588. If you practice that for one whole lesson before embarking on the actual...

Remember, I have created a list of the best online activities, games, tutorials, etc. for this (and other topics as well). All kids love games, and with multiplication tables, it's one way to give them more practice. The complete list of online activities is here , but I'll copy and paste a sample to this blogpost: Multiplication grid Drag the scrambled answer tiles into the right places in the grid as fast as you can! Multiplication.com Strategies, worksheets, games just for times tables. The Times Tables at Resourceroom.net Fill in the multiplication chart - partially or the whole thing - or take quizzes, and get graded. Explore the multiplication table This applet visualizes multiplication as a rectangle. Table Mountain Climb the mountain with 20 questions from a selected table. Multiplication table Challenge 100 questions, timed. Multiplication mystery Drag the answer tiles to right places in the grid as they are given, and a picture is reve...

Someone asked, how can you use models to multiply decimals? Learning to multiply decimals, I feel, is built on students' previous understanding of multiplying whole numbers and fractions. So models wouldn't necessarily be the focus, but instead relating decimals to fractions first, and learning from that. Multiply a decimal by a whole number Of course, when multiplying a decimal by a whole number, you could use the same models as for fractions: say you have a problem 2 × 0.34 You can use little hundredths cubes, or draw something that's divided to 100 parts. BUT you can also just use fractions, and justify the calculation that way: 2 × 0.34 = 2 × 34/100 = 68/100 = 0.68. OR you can explain it as repeated addition: 2 × 0.34 = 0.34 + 0.34 = 0.68. I employ that idea in these lessons: Multiply mentally decimals that have tenths and Multiply decimals that have hundredths Multiply a decimal by a decimal When students are learning to multiply a decimal by a decimal, th...

Well today I hopefully have something for everybody. The site DoubleDivision.org shows you an alternative long division algorithm, which takes the guessing away from estimating how many times the divisor goes into what needs divided. Also called 1-2-4-8 division. This is a pretty cool way of dividing! The interactive tool shows you the steps right there for any problem you might come up with. At MathLogarithms.com you can download an ebook by Dan Umbarger explaining logarithm how's, why's, and wherefore's in all detail for students. Great resource for precalculus students. You might also enjoy an alternative way to multiply called lattice multiplication . I did! It seems pretty simple. And lastly, if the math topics didn't interest you, how about my hubby's newest website called Laser-TVs.net ... It's about a totally new way of making TVs using lasers.

If you enjoy math 'tricks', here's one presented nicely in a slide show: How to multiply numbers with 11-19 in less than 5 seconds It explains how to multiply by 11, 12, etc. And... here's a challenge (you can ask this of students, too): After watching the slideshow, explain how it actually has very similar things going on as the standard algorithm. In other words, how do the standard multiplication algorithm (in columns) and this one compare?