Only 36% of students were able to answer the question on the right. Whereas 75% of students were able to correctly answer the problem on the left. (7) Why?
What's the big difference?
Students are more likely to relate values between objects (left question) than within an object (right question). (7)
A similar issue comes up in the questions below. 91% getting the bottom question correct, relating 11 people to 33 people. Whereas only 51% answered the top question correctly.
Students often struggle to see all of the multiplicative links between and within values.
One of the misconceptions my own students had with the right 'L' question was that the answer was 45. They had added 13cm because 8 + 13 = 21 on the base of the 'L'.
"Young children tend to see multiplication additively" Dietmar Kuchemann
DfE suggests teaching multiplication as repeated addition, with arrays, in Yr2. Yr3 scaling is introduced but after that isn't mentioned again. It is assumed multiplication is then understood and advice focusses on calculation.
Unfortunately repeated addition starts to break when answering questions like below:
Repeated addition struggles to demonstrate the multiplicative relationship with curved lines.
The above elastic band task can be further used to draw out misconceptions with repeated addition. And if needed, can be demonstrated in class using an actual elastic band.
To support multiplicative understanding we can use ratios.
Why Ratios?
Ratios are linked to representations of multiplicative relations and comparisons between quantities, such as fractions, decimals, percentages, and other visualizations of proportion such as slope. Considering ratio alongside these as overlapping lenses of proportional reasoning can be helpful. (3)
Ratio tables are representations that imply a multiplicative relationship, suggesting a calculation of multiplying or dividing parts by one another, and connected to ideas of scaling, equivalence and similarity. (2) Ratio tables are a convenient way of supporting strategies with number to obtain a solution e.g doubling, halving, multiplying by 10 etc. (10)
For example, when answering the questions below, we could write:
It is suggested that it is sequenced from using a beaded necklace for counting, to a blank number line to support addition and subtraction. A double number line is then used to support fraction, percentage and ratio problems. (5) I would add a final step to this to use ratio tables as a way of incorporating the concept from double number lines in a more concise way. It is important to realize though that while ratio tables may be more efficient, some of the structure may be lost in the compression.
Please don't worry though, if your students haven't been on this progression, they can still make productive use of ratio tables and double number lines for multiplication as scaling even when they have had little or no prior experience of using it. (5)
Ratio tables do have an advantage over double number lines as it is easier to cross multiply using a ratio tables. As Jo Morgan says in her blog post (8) it is another relationship that can be used to help solve tricky questions, in particular with algebra that double number lines may not easily be able to show. 
In the next series of blog posts I will demonstrate a number of ways you can use them to teach different topics.
Sources of inspiration:
1) https://www.cambridgemaths.org/Images/espresso_36_developing_concepts_of_ratio.pdf
2) https://www.ejmste.com/article/primary-teachers-knowledge-for-teaching-ratio-and-proportion-in-mathematics-the-case-of-indonesia-4394
3) https://www.sciencedirect.com/science/article/abs/pii/S0732312303000233?via%3Dihub
4) Beck, P. S., Eames, C. L., Cullen, C. J., Barrett, J. E., Clements, D. H., & Sarama, J. (2014). Linking Children's Knowledge of Length Measurement to Their Use of Double Number Lines. North American Chapter of the International Group for the Psychology of Mathematics Education.
5) Küchemann, D., Hodgen, J., & Brown, M. (2011). Using the double number line to model multiplication. In Proceedings of the Seventh Congress of the European Society for Research in Mathematics Education (CERME7) (pp. 326-335). Poland: University of Rzeszów.
6) Darr, C., & Fisher, J. (2004, November). Self-regulated learning in the mathematics class. In NZARE Conference, Turning the Kaleidoscope, Wellington (pp. 24-26).
7) Küchemann, D., Hodgen, J., & Brown, M. (2014). The use of alternative double number lines as models of ratio tasks and as models for ratio relations and scaling.
8) https://www.resourceaholic.com/2020/02/the-rule-of-three.html?m=1&s=03
9) https://twitter.com/mcwardgow/status/1499080227972079628?t=9RJfyFQx-CUx5DYMhsTnLQ&s=19
10) Dole, S. (2008). Ratio tables to promote proportional reasoning in the primary classroom. Australian Primary Mathematics Classroom, 13(2), 18-22.