Updated: Apr 28
Multiplication and Division Models
At Mentorhood, we love visuals. We’re not the only ones to use visuals to introduce the concepts of multiplication and division; it’s not a-typical to see small pictures circled into groups of different sizes, or partitioned off into columns and rows. This helps new learners understand the relationships between totals and groupings, and between multiplication and division themselves.
Most students, when they are asked to repeat this foundational understanding into a word problem scenario, they are expected to tease out from the words which operation they are using. They can then use the visual aids, if they’d like, to help them break down the arithmetic, but they are expected pretty quickly to deviate from a visual approach into a symbolic one.
Our bar model approach enables students to preserve the understanding they cultivated in earlier stages of learning, such as when they represented relationships between numbers visually in addition and subtraction scenarios. Bar models don’t just break down of operations. Mentorhood’s Singapore Math approach ties a visualized component to terms such as “groups of” and “times as many as” within the context of a larger question.
Bar models also force students to represent all the information given to them in the question and to leave holes where variables are missing. While with simple questions, a student may simply be able to glance at the question, pull out two given numbers, and form a correct number sentence, the skill of accounting for all numbers becomes crucial when questions become more complex. Bar models help students avoid a situation of simply assuming the most obvious answer, while ignoring a subtle piece of information or an additional required step.
The Power of a Visual Approach
Bar models help the students answer “Yes” to the following questions:
1) Can you figure out what you know (what is given in the question)?
2) Can you figure out how that information fits together in relationship?
3) Can you identify what you are missing?
4) Can you pinpoint exactly which pieces of information you need to get your final answer?
5) Can you fill in those blanks by manipulating the numbers you are given to form new conclusions?
Let’s see how bar models are used to introduce the simplest multiplication and division grouping questions.
Whole-Part Multiplication and Division
Recall that bar model questions can reflect a whole-part scenario or a comparative relationship, depending on whether there is more than one subject. We start students off with whole-part situations, as they typically represent a single total made up of multiple groups of an equal amount and involve only one step.
WHOLE-PART COMPARISON MODELSMODELS
Whether multiplication or division, there will be a single bar divided into multiple parts. With a simple multiplication question, a student will typically be given the number of groups and the number in each group and will be missing the total.
With a division question, they are given the total and solving for a component; however, there is an extra consideration. The student will be given the total and either the number of groups or the number in each group. If they do not know the number of groups, they can represent the missing groups with dotted lines. Either way, the arithmetic is the same, but it’s important for students to understand exactly what they are solving for.
Comparative Multiplication and Division Relationships
When introducing a comparative component, the bar models will look relatively similar for multiplication and division, again with the possibility that a division question will have dotted lines representing an unknown number of groups. As with whole-part models, the key difference between a multiplication question and a division question will lie in which variable is missing: a total (multiplication) or a component (division).
The key with comparison models in multiplication and division questions is that the subject with the larger amount will have a box that is a certain number “times as big” as the other subject, represented by a number of equal-sized boxes.
As students become more advanced, they are able to handle questions that blend both multiplicative and divisive requirements. Sometimes, questions will even use both whole-part and comparison ideas.
Especially in comparison questions, when one subject’s total is a certain number “times as many” as the other subject, the visual representation of the equal-sized boxes can unlock a ton of calculative potential. As in the list of questions above that a successful bar model student is able to answer “Yes” to, showing what parts of two numbers can be equalized, broken down, or scaled is key to being able to fill in the blanks left by the question. Here is an example of when that power becomes critical:
This question would be immensely confusing without a second-nature understanding of how to break down multiplication and division scenarios into visualizable pieces that relate to one another. Students can deduce from a properly executed model exactly how to use each operation to gain the next piece of information they need to solve their question.
Patterns of Thought
At Mentorhood, we believe strongly in bar models. Even for the oldest children, Singapore-style bar models help students apply new concepts on top of strong foundations, enabling them to mentally visualize where their answer will be coming from.
Once they are mastered, bar models simplify nearly impossible questions into fill-in-the-blank puzzles. We aren’t just teaching how to understand operations; we are teaching how to find patterns and use them to one’s advantage.
In today’s competitive and ever-changing world, we treasure the ability to give even the slightest advantage to our students. We’re thrilled that you’ve chosen to investigate for yourself the Mentorhood Math advantage.
Welcome to the team, parents—the Singapore Math team! We’re excited to have you on board and we look forward to helping you coach your little ones.
We’re happy to come alongside your learner in their learning journey. And if you want to do more from home, you can check out our ever-growing library of games and courses, now on YouTube!
Specifically, we’ve been working through a course on bar models. Check it out below!
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