Interpret multiplication as scaling (resizing), by:
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Multiplication can be thought of as resizing a quantity. Multiplying 12 by 3 stretches it to three times its size, while multiplying 12 by 1/3 shrinks it to a third of its size. Fifth graders learn to predict the size of a product compared with one factor just by looking at the other factor, without calculating.
The rule of thumb comes from the scale factor. Multiplying by a number greater than 1 makes the result bigger than the starting number, multiplying by a number less than 1 (but more than 0) makes it smaller, and multiplying by exactly 1 leaves it unchanged. This last case explains fraction equivalence: multiplying 3/4 by 2/2 is multiplying by 1, so 6/8 must have the same value as 3/4. Students who reason this way can tell at a glance that 7/8 × 245 is a little less than 245, and that 5/4 × 9 is more than 9.
Students carry this belief from whole numbers. Showing that 1/2 × 10 = 5 with a strip diagram challenges it directly.
Some students think 5/4 is small because 4 is in the denominator. Comparing the numerator with the denominator tells whether a fraction is more or less than 1.
On questions designed for estimation, students who compute every product waste time and may make errors. Train them to decide first whether the scale factor is above or below 1.
Fractions such as 4/4 or 7/7 are equal to 1, so multiplying by them does not change the size. Students should spot these before comparing.
Without calculating, order these products from least to greatest: 3/5 × 40, 9/9 × 40, 7/4 × 40.
Answer: 3/5 × 40, then 9/9 × 40, then 7/4 × 40.
Use a stretchy band or a resizing picture to make 'scaling' concrete. Number lines are also effective: mark a number, then show where 1/2, 1 and 3/2 times that number fall.
Typical items ask which product is greatest, whether a product will be more or less than a given factor, or ask students to explain the effect of multiplying by a fraction.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: A) It is less than 318
5/6 is less than 1, so multiplying by it makes the number smaller.
Answer: D) 4/3 × 25
4/3 is the only factor greater than 1, so 4/3 × 25 is the only product greater than 25.
Answer: 27
3/2 × 18 = (3 × 18)/2 = 54/2 = 27, which is greater than 18 because 3/2 is greater than 1.
Answer: B) Because 2/3 × 4/4 multiplies by 1, which does not change the value
4/4 = 1, and multiplying any number by 1 leaves it the same size, so 2/3 × 4/4 = 8/12 has the same value as 2/3.
Answer: 6 (also accepted: 6 cm)
3/5 × 10 = 30/5 = 6 cm, smaller than 10 because 3/5 is less than 1.
A full lesson with slides, activities and an exit ticket on multiplication as scaling, pitched to grade 5 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 5.NF.B.5 with an answer key, ready in about a minute.
Make a worksheet →Turn multiplication as scaling into a quiz students answer online that marks itself, with a class summary for you.
Build a test →It means thinking of multiplication as resizing: a factor greater than 1 stretches a quantity, a factor less than 1 shrinks it, and a factor of 1 keeps it the same.
The core skill is predicting the relative size of a product without calculating. Calculating afterwards is a good way to check the prediction.