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Chapter 13: Problem 1

Draw pictures showing how \(57 \times 4\) could be modeled: with counters, with base-ten blocks, with arrays, or with areas on base-ten grids.

Short Answer

Expert verified
Model 57 脳 4 using counters, base-ten blocks, arrays, or grid areas for a visual representation.

Step by step solution

01

Understand the Multiplication Problem

We need to model the multiplication of 57 by 4 using different visual methods such as counters, base-ten blocks, arrays, or areas on base-ten grids.
02

Model with Counters

Using counters, represent the number 57 by grouping counters. For example, use 57 counters in total. Then, repeat this grouping 4 times to model the multiplication.
03

Model with Base-Ten Blocks

Use base-ten blocks to represent the number 57 as 5 blocks of 'tens' and 7 blocks of 'ones'. Then, replicate this arrangement 4 times to account for multiplication by 4.
04

Model with Arrays

Create an array with 57 rows, each having 4 columns. This represents arranging the number 57 in groups of 4, visualizing the multiplication process.
05

Model with Base-Ten Grid Areas

On a base-ten grid, draw a rectangle representing 57 units horizontally and 4 units vertically. The area of the rectangle gives the result of the multiplication.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Base-Ten Blocks
Base-ten blocks are a fantastic visual tool to help understand multiplication. Imagine each block as a "ten" unit and each small cube as a "one" unit. So, when you're multiplying 57 by 4, start by breaking 57 into 5 tens and 7 ones.
To visualize this with base-ten blocks:
  • Take 5 ten blocks to represent 50.
  • Add 7 single units to make up 57.
Then, replicate this setup 4 times to model multiplication. Each grouping gives a visual sense of what 57 multiplied by 4 looks like. It helps to see how repeatedly adding sets of 57 results in a larger sum. This method demonstrates multiplication by breaking numbers into manageable parts, showing repeated addition clearly.
Arrays
Arrays are a great way to visualize multiplication problems. You create arrays by organizing objects into rows and columns. Think of it as a grid-like arrangement.
For 57 times 4, you create an array that has 57 rows and 4 columns. Each row represents a single group of 4.
  • The columns represent the multiplier (4 in this case).
  • The rows represent the multiplicand (57).
From this arrangement, you can easily see the repeated addition of 4, making it simple to understand how multiplication works. This visualization supports the idea of distributing a number into equal parts and gives a tangible form to abstract arithmetic operations.
Counters
Counters make learning multiplication straightforward by showing it as a repeated addition of groups. Imagine using small objects like coins or discs as counters to represent numbers. For 57 imes 4:
  • Start by gathering 57 counters.
  • Repeat the set of 57 counters 4 times.
This way, you replicate the number 57 four times, exhibiting repeated addition. Each group of 57 is a "layer" of multiplication. This technique lets you visually observe the entire multiplication concept through tangible items, which can be arranged and counted easily to find the total.
Area Model
The area model uses the concept of rectangles to represent multiplication. It connects abstract numbers with real, visual shapes. To use the area model for multiplying 57 by 4:
  • Draw a rectangle.
  • One side of the rectangle should measure 57 units (length).
  • The other side should be 4 units (width).
The area of the rectangle, calculated by multiplying the length and width, gives the product. This model helps visualize multiplication as filling up space, showing how the total area (or product) grows as you add more layers or rows to the grid. It's particularly useful in understanding what happens when numbers are multiplied, enhancing comprehension by displaying the multiplication process spatially.

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Most popular questions from this chapter

Which division concept, measurement or partition, is easier for direct modeling and is also the one used to develop the standard long-division algorithm? Make up an appropriate word problem to go with \(735 \div 6\).

What would you do if your students seemed to persist in using repeated addition for multiplication problems without really doing any multiplication?
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