by C. Elkins, OK Math and Reading Lady
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
by C. Elkins, OK Math and Reading Lady
Doubling and . . .
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.
3 x 4 = 3 x (2 x 2) which equals (3 x 2) x 2
5 x 12 = 5 x (2 x 6) which equals (5 x 2) x 6
by Cindy Elkins, OK Math and Reading Lady
For the teachers in OK, there have been many changes to the math standards, called OAS (Oklahoma Academic Standards). For 3rd, 4th, and 5th grades there is a stronger (or new) emphasis on growing patterns in the algebraic reasoning section. As I always advocate, connecting this concept to concrete objects and pictorial representations should happen before students are given a string of numbers and a function table to decipher.
Did you know there is more than one type of pattern? Continue reading