Math
Multiplication
Multiplication means "equal groups of..." So when we read 6 x 7 = 42, we say "Six groups of seven is 42." This is important to emphasize the multiplication symbol means "groups of" as students develop their conceptual understanding of multiplication.
Concrete --->Visual/Pictorial ---> Abstract
When learning a new concept, it's important to begin with concrete objects before moving to visual representations of the objects. In multiplication students built arrays or made groups of objects. Then they moved on to drawing pictures of arrays or using open arrays and number lines. The last step is abstract, which means just using numbers. If your child is having difficulty go back to concrete or visual strategies of solving a problem.
What is multiplication?.MOVWhat is multiplication? Here are four representations of multiplication we use in grade 4.
Fluency Without Fear: "Mathematics facts are important but the memorization of math facts through times table repetition, practice and timed testing is unnecessary and damaging...It is useful to hold some math facts in memory. I don’t stop and think about the answer to 8 plus 4, because I know that math fact. But I learned math facts through using them in different mathematical situations, not by practicing them and being tested on them." (Jo Boaler)
FluencyWithoutFear-2015-1.pdfCheck out this article for a more in depth understanding of math fact fluency.
Repeated Addition Multiplication Strategy.MOVRepeated Addition Strategy...because multiplication is fast addition
Array and Split the Array Multiplication Strategies.MOVArray/Area Model and Decompose the Array Strategies (Visual Strategy)
IMG_2009.MOVA student explains how to decompose or split an array to make the multiplication easier.
By decomposing this array, the student has made 18 x 6 easier to solve. She decomposed the 18 into a 5, 5, 5 and a 3.
She then multiplied each by 6. She added up all the decomposed parts of the array and got 108 cupcakes.
Number Line Multiplication Strategy.MOVOpen Number Line Strategy (Visual Strategy)
This strategy shows how repeated addition works, however it can become a very efficient strategy even for 2-digit multiplication. Check out these video. -------->
IMG_4883.MOVIMG_4884.MOVIMG_4885.MOVDecomposition Using an Open Array
The open array is far more efficient than an array. A rectangle is drawn to visualize the problem.
This student used an open array to solve 8x25. She saw two 4s when she decomposed the 8. This made it easier for her to multiply.
(4x25) + (4x25) = 200.
Another student decomposed the 25 to make the problem easier. She saw two 10s, a 3 and a 2 inside 25.
(10x8) + (10x8) + (3x8) + (2x8) = 200
This student solved 24x7 by decomposing the 24 into easier numbers to multiply. She saw a 10, two 5s and two 2s in 24.
Solving 57x4 is made easier by decomposing the 57 into a 50 and a 7 and then multiplying them by 4. Add up the parts (200+28) = 228.
Using an Open Array in a Problem
This student used open arrays to decompose two digit numbers that were more difficult to multiply.
IMG_2551.MOVWatch how this student uses open arrays to decide which amount of money she would want.
Double and Halve strategy .MOVListen to and watch students explain how the doubling and halving strategy works...and when it doesn't.
Double and Halve
Compensation (Using a Friendly Number)
This strategy works well when one of the factors is close to a friendly number. For example, 5 x 19 (5 x 20), 28 x 4 (30 x 4) or 99 x 8 (100 x 8).
Take a look at some examples of compensation in these videos.
This student used a parts-whole model to show the meaning of her story problem. She then used compensation (top right hand corner) to multiply 24x4. Instead of solving for 24 groups of 4, she solved for 25 groups of 4 because it was easier. She then took 1 group of 4 away from the total.
IMG_4886.MOVIMG_4887.MOVDecomposition: this is a more abstract strategy than the visual strategies of arrays and number lines.
IMG_4938.MOV