As I promised, here is a post about another multiplication and division story structure.
The dinosaur is about twice as tall as the human. The dinosaur is about 12 times wider than the human.
While the previous structure I mentioned dealt with equal groups, this one deals with comparisons. Like with addition / subtraction comparison problems, I will show some ways to use double bars to illustrate these. (I just added a FREEBIE – a 2-pg. pdf showing the pictures of the problems shown below – get it now Equal groups comparison problem pics)
Notice some of the questions associated with these stories. They sound a lot like questions used in other story structures — this is why I advise against statements like, “If it asks how many in all, add the numbers together.” Notice how this method guides a student as they go from pictorial to abstract. You can also represent these types of problems with cubes or money manipulatives if you need a concrete example.
This problem shows a multiplying process.
Read the problem. Ask who and what this story is about (Joe and Brent – Joe has more $ than Brent). Notice there is no additive or subtractive process, but one has more – the other has less. This is a signal to use comparison bars to help solve.
Make a vertical line (to help line up the bars evenly on the left side. Draw the bar with the known amount. (Joe has $22.) Label the second bar (Brent).
3. Since the story said Brent has 3 time more, draw 3 bars equal to the size of Joe’s bars.
4. Since Joe’s bar is $22 and the bars by Brent’s name are equal in size, all of Brent’s bars should be labeled with $22.
5. To find out how much Brent has, solve by repeated addition or multiplication.
6. IF the question was slightly different (How much do Joe and Brent have all together?), then the above steps would have to be followed by 1 more step (adding the two amounts together).
* If students had been taught earlier to “Add when the question asks How many in all” then the child would likely add $22 + 3 to get their answer. That signals no conceptual understanding of what the problem is all about: One has more, the other has less.
Are you ready for a way to help students solve multiplication and division word problems? I have developed a template that will help students record the given facts and think through the process involving equal groups. There are 3 basic types of equal groups problems:
# of groups and # in each group are known
# of groups and total are known
# in each group and total are known
Pictured here is an illustration of the strategy which can be used for multiplication and division problems. Get my FREE packet right here: Equal groups strategy with template.
One of the most important steps I recommend when working with equal groups problems is for students to brainstorm things that typically come in equal groups. There is a great book titled “What Comes in 2’s, 3’s, and 4’s” which can set the stage. It’s a picture book, but helpful to get kids’ brains warmed up. Here are some visuals I made to illustrate the point. Click here for your FREE copy. They are included in this set of Equal groups pictures and list template
Kids may need your help to think of things to add to the list. See some hints below. This template and a full list is included in the above FREE download. Continue reading →
Is division a dreaded topic on your list of objectives to teach? Like many math topics, students have a harder time understanding it most likely because it’s not something they use regularly in their lives. Students should understand why division is useful before they have to start solving division problems. In this post, I will focus on helping students see the relationship between subtraction, multiplication, and division both with concrete objects, pictures, and the partial quotients model. Freebies available below!!
425 divided by 8 picture form
425 divided by 8 partial quotients model
Then let’s talk about what division really is — it is repeated subtraction; much the way multiplication is repeated addition. The issue is that repeated subtraction is not always very efficient. Here’s what I mean.
Let’s say I have the basic problem 25 ÷ 5. I could start with 25 and then subtract 5, subtract another 5, another 5, another 5, and another 5 until I run out and reach zero. I would have to do this 5 times. If I had 25 cookies that I wanted to share equally among 5 friends, I could do the “one for you, one for you, one for you, one for you, and one for you” process and still end up with 5 cookies for each. Or I could try “two for you, two for you,” etc. to make the action of passing out the cookies faster. When I get down to 5 cookies, I return to the “one for you . . .” to make it work.
With a larger problem such as 72 ÷ 6, I can again try subtracting 6 at a time until I reach zero. This would take 12 repetitions — not efficient, but still accurate. Could I subtract 12 at a time instead (2 groups of 6) to be more efficient? Or 18 at a time, or 24 at at time? This is the idea behind the partial quotients model I will refer to a little later. Continue reading →
This is the last part of number operations dealing with decimals: multiplication and division in a concrete and pictorial method. This is actually not part of the KG-5th grade OAS standards, but it is addressed in 6th grade and for those of you utilizing the CCSS, you will find multiplication and division with decimals starts in 5th grade. Or you may have advanced students who are ready to explore this concept. There are a couple of freebies included in this post. Read on to find them!
Typically we teach our students that when you multiply 2 numbers together, the product is larger than the 2 factors. And when we divide two numbers, the quotient is smaller than the dividend.
Be careful about stating this generalization: This remains true when multiplying whole numbers (or even a combination of whole and decimal such as 5.2 x 6.4 = 33.28 in which the product is larger than either factor), but NOT with decimals or fractions less than 1 (example: .7 x .2 = .14 in which .14 is less than either factor .7 or .2). With division of decimals less than 1, the answer is often a whole number larger than either the divisor or the dividend (such as 5 divided by 1/2 = 10). This is a difficult concept, but modeling and practicing with concrete and pictorial models helps to see the reasoning. So, be careful not to say, “When you multiply two numbers together you always get a bigger number.” because it’s not always true.
To further understand this principal, let’s look at the use of the times (x) sign with whole numbers. The problem 5 x 2 could be shown in an array as 5 rows with 2 in each row (phrased as 5 “rows of” 2). It can be shown in a set model such as 5 groups and each group has 2 (phrased as 5 “groups of” 2) . It can also be shown in an area model (box) as a shape divided equally into 5 rows and 2 columns (phrased as 5 “by” 2). Continue reading →
One of my subscribers really liked the multiplication strategies I posted in my 4 part series, but was finding it difficult to copy them for easy reference. So . . . I pooled pictures of the boards onto a single PDF (it’s 10 pages) and am providing it here (and also with the 4th part of the series). Click here for your copy: Multiplication Strategies PDF
I hope you have had a chance to look at Parts 1, 2, and 3 of my multiplication strategy posts. These strategies are especially helpful with 3rd – 5th grade students (and beyond). I have been reading a book by Dr. Nicki Newton called “Guided Math in Action.” She discusses five components to being mathematically proficient. One of them is strategic competence. What is strategic competence? The National Research Council defines it as “the ability to formulate mathematical problems, represent them, and solve them.” The first process std. in Common Core (Make Sense of Problems and Persevere in Solving Them) emphasizes strategic competence in this way: “they try special cases and simpler
Halving Multip. Strategy
forms of the original problem in order to gain insight into its solution, . . . students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? . . . and, they can understand the approaches of others to solving complex problems and identify correspondences between different approaches.”
With all of that said, I would like to show two other multiplication strategies: 1) Doubling and Halving, and 2) the Lattice Method. All of the strategies picture boards I have shown are available on this free PDF (It’s 10 pages): Multiplication Strategies PDF
Doubling and Halving Doubling and halving is supported by the associative property. It also enables the student to use mental math strategies. Here is an example: Original Problem: 25 x 12. When changed to 50 x 6, I can solve it mentally which equals 300. From what I have researched and applied, here are some tips:
Use when a problem has one even and one odd factor (or two even factors). It won’t work with 2 odd factors because you can’t break an odd number in half and still get a whole number.
Double the odd factor and halve the even factor. Do this 1-3 times until you get two factors which can be multiplied mentally.
Most useful when the odd factor has a 5 in the one’s place (because 5 doubled is 10, so the result will be a multiple of 10 which can usually be taken care of mentally).
How is the above problem connected to the associative property?
Think of 25 x12 as 25 x (2 x 6).
If I apply the associative property, I get (25 x 2) x 6 = 50 x 6 = 300.
I can also go another step further and show that 50 x 6 = 50 x (2 x 3) = (50 x 2) x 3 = 100 x 3 = 300.
Here is another one: Think of 15 x 24 as 15 x (2 x 12) = (15 x 2) x 12 = 30 x 12 = 360.
See my pictures below of how that looks in array form with the problems 3 x 4 and 5 x 12.
In Multiplication, Part 3 I will focus on 3 strategies for double digit numbers: area model, partial products, and the bowtie method. Please also refer back to my Dec. 6th post on Number Talks for 3rd-5th grade where I mentioned these and other basic strategies for multiplication. I highly recommend helping students learn these methods BEFORE the standard algorithm because it is highly linked to number sense and place value. With these methods, students should see the magnitude of the number and increase their understanding of estimation and the ability to determine the reasonableness of their answer. Then, when they are very versed with these methods, learn the standard algorithm and compare side by side to see how they all have the same information, but in different format. Students then have a choice of how to solve. Try my “Choose 3 Ways” work mat as bell work or ticket in the door. Get it free here.
Area Model: This method can be illustrated with base ten manipulatives for a concrete experience. Remember the best methods for student learning (CPA) progresses from concrete (manipulatives) to pictorial (drawings, templates, pictures) to abstract (numbers only). Using a frame for a multiplication table, show the two factors on each corner (see examples below for 60 x 5 and 12 x 13). Then fill in the inside of the frame with base ten pieces that match the size of the factors. You must end up making a complete square or rectangle. This makes it relatively easy to see and count the parts: 60 x 3 and 5 x 3 for the first problem and (10 x 10) + (3 x 10) + (2 x 10) + (2 x 3) for the second. I’ve included a larger problem (65 x 34) in case you are curious what that looks like. The first 2 could be managed by students with materials you have in class, but I doubt you want to tackle the last one with individual students – nor do you probably have that many base ten pieces. A drawing or model would be preferred in that case. The point of the visual example is then to connect to the boxed method of the area model, which I have shown in blank form in the examples . . . and with pictures below. I also included a photo from another good strategy I saw on google images (sorry, I don’t know the author) which also shows 12 x 13 using graph paper. Continue reading →
In part 2, I will show you some ways to help students decompose a multiplication problem into 2 (or more) easier multiplication problems. Most students know problems with factors of 2, 5, and 10. The decomposing will allow students to use what they know to work on the unknown / unmemorized fact.
I frequently see students struggle with solving an unknown multiplication problem. Often they choose skip counting, but if they miss just one number in the sequence, the answer comes out wrong. I also see them use their multiplication chart, but this doesn’t do much to help them apply number sense. Other times I see students draw circles with dots inside, but this is time consuming and it often becomes just a counting practice. This method is using the distributive property. Students can break apart one of the factors into “friendly” addends. I usually advise making one of the addends a 2, 5, or 10 since those are usually easier to compute or are already memorized. Here are some examples:
Are you looking for some ways to help your students learn the multiplication facts? Or ways to help them solve multiplication problems while they are in the process of learning the facts? One way is to skip count or repeatedly add the number over and over again. While this is one acceptable strategy, I see many students skip count using their fingers, often starting over numerous times. And if the child miscounts just one number in the sequence, then all of the remaining multiples/products are incorrect. Sometimes the student will write down the sequence in a horizontal row (better than using fingers in my opinion), but again – if they miss one number . . . all the rest of the numbers in their list are wrong.
What I want to show you today (in Part I of my series about multiplication strategies) are ways to relate the multiples/products in recognizable patterns which may facilitate recall and help with committing the facts to memory. Yes, students should also know the following about multiplication:
Multiplication is repeated addition. For example: 3 x 4 means 3 groups of 4 or 4 + 4 + 4 = 12
Multiplication is equal groups. 3 x 4 might be shown with 3 circles and 4 dots in each one. Be cautious about continued use of this one. Students are good at drawing this out, but then are they actually adding repeated groups or just counting one dot at a time. Observe students to see what they are doing. Transition to showing 3 circles with the number 4 in each one.
Multiplication is commutative. If solving 7 x 2 (7 groups of 2), does the student count by 2’s seven times, or perhaps make it more efficient by changing it around to make it 2 x 7 (2 groups of 7 — and adding 7 + 7)?
Multiplication can be shown with arrays. If students are drawing arrays to help solve, watch how they are computing the product. Are they counting one dot at a time? Or are they grouping some rows or columns together to make this method more efficient. I will focus on this one in Part 2.
Multiplication can be shown by skip counting.
I have included 2 visuals to see some of my favorite ways to relate skip counting to unique patterns (for the 2’s, 3’s, 4’s, 6’s, 8’s, and 9’s). And another pattern below:
An even number x an even number = an even number
An odd number x an even number = an even number
An odd number x an odd number = an odd number
Skip counting by 2, 3, 4
Skip counting by 6, 8, 9
Next post will be Part 2 of Multiplication Strategies. Have a great week!
This is the Part 3 of Number Talks. If you are just tuning in, please refer to NT Parts 1 and 2. As I mentioned before, conducting a Number Talk session with your students is a chance for them to explain different ways to solve the same problem. This is meant to highlight strategies which have already been taught.
Click below to watch 2 videos of how to conduct a Number Talk session with intermediate students. You will see many strategies being used.
Addition and Subtraction Strategies: I like using the methods listed below before teaching the standard algorithm. This is because they build on a solid knowledge of place value (and number bonds 1-10). If your students are adding and subtracting using the standard algorithm and can’t adequately explain the meaning of the regrouping process in terms of place value, then try one of the following methods. In many cases, I will ask a student the meaning of the “1” that has been “carried” over in double-digit addition. About 85% of the time, the student cannot explain that the “1” represents a group of 10. When adding the tens’ column, they often forget they are adding groups of 10 and not single digits. So they get caught up in the steps and don’t always think about the magnitude of the number (which is part of number sense). You will notice teachers write the problems horizontally in order to elicit the most strategies possible.
Place Value Decomposition
Open Number Line (to add or subtract)
Here are some possible Number Talk problems and solutions:
Notice use of place value and number bonds.
These methds build strong number sense.
I call this “Facts Of.” You can use any number.
Multiplication and Division Strategies: I like using these methods before teaching the standard algorithms. Again, they build a solid understanding of place value, the use of the distributive property, and how knowledge of doubling and halving increases the ability to compute problems mentally. Once these methods have been learned, then it is easy to explain the steps in the standard algorithm.
Doubling and Halving
Here are some possible Number Talk problems and solutions:
Use the known (6 x 2) to learn the unknown problem.
Use the distributive property!
To divide by 4, halve the number twice. To divide by 8, halve the number 3 times.