Decimal multiplication: part of an amount
Interpret decimal multiplication as repetition or a fraction of an amount.
Begin with the activity ↓Can multiplication make an amount smaller?
Multiplying a positive amount by a number below one takes a part of it. Multiplication does not always make it larger.
Joining three 0.6 m lengths combines 0.6 three times. Six tenths repeated three times give eighteen tenths, or 1.8 m. The whole-number idea of equal groups still works.
Now take 0.4 of 0.6. Since 0.4 is four tenths, divide 0.6 into ten equal parts and take four of them. There is no reason for the result to exceed 0.6.
Divide a unit rectangle into ten columns and ten rows. Six columns overlap four rows in 24 cells. That is 24/100 of the original whole, or 0.24.
Calculating only 6×4=24 loses the units. Since 0.6=6/10 and 0.4=4/10, their product is 24/100. Counting decimal places abbreviates this multiplication of smaller units.
Taking 0.2 of 2.5 means taking a fifth of it, giving 0.5. Twice 2.5 is 5. Whether the multiplier lies below or above one helps you check the size of a positive result.
Keep track of what the result measures. Scaling 0.6 m by 0.4 gives a length of 0.24 m. This differs from multiplying two lengths to find an area: here the second number is a scale factor.
Listen one paragraph at a time
Hold on to this distinction
In 0.6×0.4, the 24 counts hundredths. Taking only 0.4 of the starting amount also explains why the positive result shrinks.
Change it and check
Predict whether the result will grow or shrink. Then compare the fraction calculation and two bars using a common scale.
0.6 × 0.4
Choose another multiplication expression above and compare the amounts. A zero starting amount stays zero under any displayed multiplier.
We multiply nonnegative decimals by unitless scale factors. Both bars share a display scale; they are not physical rulers.
Try with paper and objects
Draw a ten-by-ten square. Hatch six columns vertically and four rows horizontally, then count the overlaps.
Check and explain another example
Check your idea 1
Check your idea 2
Apply it in a new setting
Connect the answer to its unit
Return to the explanationCompare with a solution
0.24 · m
8/10×3/10=24/100=0.24.
Does your explanation include these ideas?
Explain aloud or in a drawing, then check your ideas. Different wording can express the same meaning.
Continue, or pause here
Understand why changing both quantities by the same factor preserves their quotient.