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Problem:

69 X 4 = (70 X ?) - (1 X ?)

This is our VERY first day learning the distributive property.    (Brand new for child.)   I *know* the answer is 4, but I am having problems explaining why it is 4 in a way that my child can understand.   Can anyone give me some good language to use?

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You are counting by 4s, but oops, you counted 1 too many and have to take a 4 away.

I'd explain it first with a smaller number and it helps if they've done problems like 39+17= 40+17-1.

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Maybe start with a smaller problem that you can model with manipulatives. Some number times 10 vs sum number times 9 so they can see the difference. So, if you pick three, what's the difference between 10 groups of 3 vs 9 groups of three. It will be easier to see that it's one group of 3. Then you can bridge to larger numbers.

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69 groups of 4 is the same as 70 groups of 4 minus 1 group of 4.  Some people find it faster to calculate mentally by looking at the groups.

If the student can calculate 69X4 mentally (and many can), he may need larger numbers to find the distributive property personally useful.

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I would think it would be easier to teach it as 69 x 4 = (60 x 4) + (9 x 4).

That's adding, and fairly intuitive because you can see the 60 and the 9 in the number 69. Then you can move on to the other version.

(Or did you already cover that?)

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How does your child figure out the answer to 9x7? If they can't remember off the top of their head they might figure out 10x7 and then subtract a group of 7.

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emphasize that 69 = (70 - 1) and that it's easier to multiply 70 than to multiply 69 in one's head.

69 x 4

= (70 - 1) x 4

then the 4 is "distributed"

= (70 x 4) - (1 x 4)

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Yes, we did with smaller numbers and with manipulatives.   They understand why the distributive property is true...but they are struggling to go "backwards" with the distributive property...... and I am having the hardest time explaining it in words!  (Which probably indicates that MY understanding is not as strong as it should be.)

In other words, they seem to get why....

(2+8) X 5 = (5X2) + (5X8)

But, they are having problems going from:

10 X 5 = (? X 2) + (? X 8)

I tried to tell that we can see that 2+8 = 10....so we know that the "x5" must be "distributed" through the 2 and 8...So the "?" would equal 5.   That is how I think of it.  But they are looking at me with blank stares.

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10 X 5 = (? X 2) + (? X 8)

You have a lot of good answers already. This problem seems a bit weird to me, in the sense that why would you want to do this problem? 5x10 is an easy multiplication and this problem is complicating it unnecessarily. How would this problem ever come up in a practical sense?

Why is the distributive property important to learn? One obvious application involves the ability to manipulate numbers more easily, which leads to easier mental math. This is evident in your first example, where 69 = 70 - 1. Then there is the ability to better understand multi-digit multiplication and place value, where you can use expanded form so 38 x 17 = (30 x 17) + (8 x 17). And then it's important for algebra as you learn to combine and separate terms in various ways.

But I find your example problem non-inutitive. Plus if I wanted to use a manipulative to show something like 2 groups of 5 and 8 groups of 5, I can't even really see how do that immediately, because of the way this problem chooses to have the 5 number missing.

What if you gave them this type of problem, but with one of the '5's filled in so there was only one blank? Would they immediately be able to see the other was 5 too? The other thing is that, on the face of it, if you know the distributive property this is a very easy problem. Maybe the idea is that if they see it several times in examples, it's supposed to be an easy one too. Oh the 2 and the 8 make 10 therefore the other number, 5, is the one that's missing? Is that the thought that the problem writer wanted them to have?

ETA: is your book even specifying (and emphasizing) that the two ?s have to be the same? Otherwise I could easily answer that 10 X 5 = (21 X 2) + (1 X 8) and it's not the distributive property at all. I really think I don't care for this question format.

I really like the idea of having manipulatives here, with 10 groups of 5 and asking, how can we divide these into two parts without disrupting the individual groups of five, and then you can see how the distributive property works more. But, again, doesn't work with the format of the question you are looking at.

Edited by debi21
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for 69x4 =  (70 x ?) - (1 x?)  you need to help them read the symbols.   You have to lead them to 69 groups of 4 is the same as 70 groups of what minus one group of what.  If they stumble on the parenthesis, or they don't quite understand an equals sign or the multiplication symbol, they won't be able to translate into words and visualize the groups and problem. They also have to realize the 'what' is the same 'what' , not two different quantities.

Edited by Heigh Ho

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For going backwards, maybe considering the area of a rectangle would help? Draw a 5 x 10 rectangle. Then the area is 5x10. Or we can cut the rectangle in two and to find the total area add the sums of the areas of the two smaller rectangles together, e.g. a 5x2 rectangle plus an 8x2 one.

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