# Must Have Manipulatives for Mathematics: Junior Kids

## Aren’t Junior Kids too Old for Manipulatives?

No one is too old for manipulatives. These thinking tools are used at all levels of instruction and in the work environment as well.

You just need to stop and think about when you had your latest car repair. The mechanic may have shown you a set of worn-out brake pads and a set of new ones to help you understand the need for a new set. Words alone do not adequately explain all of the details of the problem, but with a physical representation of your brake pads, you can clearly see where the issues lie.

There are many reasons to use manipulatives but the most important tops this list.

1) FUN The easy answer is that they are FUN. Motivation is critical for learning. The more appealing the materials, the more likely the lesson will be successful. Motivation is the best way to help with math.

2) See the Wheels go Round: Manipulatives make your child’s thinking visible. This is great for teaching, as well as for diagnosing any little glitches that can happen.

3) Link to the Real World: Some manipulatives reflect the real world. How better to learn integers than examining a thermometer that has a scale below zero? Experience financial situations through game experiences .

4) Professionals use Them: Many professionals use manipulatives regularly. Where would architects be without their basic 3D shapes? When you visit your doctor or dentist, they may use a model (which is a manipulative) to explain a procedure for you or a technique you need to do to get better.

But I probably had you with FUN.

So, if you are working with kids from 9 to 11, what are the best manipulatives for math that you need to have, and how do you use these tools effectively?

## Allow Free Exploration Time First

Usually, manipulatives are very appealing to all individuals, even adults. It is important to give exploratory time with the manipulative before you decide to use them in any organized way. Make sure you have given ample free playtime for kids and adults to build or explore the possibilities with manipulative before using them in your lesson. Some kids will need a week or more of time to fully explore these 3D objects.

Also, you may not want to pack away the manipulative when you have finished the lesson as your child may want to continue exploring mathematical concepts during their free time. Many teachers keep these objects available all day to be used for play.

## The Areas of Mathematics

Most junior math curriculum is divided into 5 sections, and the manipulatives are often very different for each type of mathematical concept: Numbers, Algebra (patterning), Data, Spatial Sense, and Financial Literacy. Woven through every activity are some social-emotional goals to maintain a positive outlook while working with mathematical concepts. The basis of the following summary comes from an article: New Math Curriculum 1 to 8.

Let’s take a look at what types of manipulates are essential for each area of mathematics. They will help with math lessons.

### Numbers

By far, when you introduce the word mathematics, most people conjure up numbers first. Kids need a multitude of experiences with numbers as numbers act as a foundational learning experience for mathematics. Junior kids begin to understand the complications of larger numbers and a variety of operations. Here are some of the milestones by grade levels in most districts. The following expectations are grossly simplified. To understand the details, you will need to refer to your area’s curriculum. They give parents a baseline for instruction but do not include any teaching strategies.

• understand and use numbers to 10,000
• continue investigations with fractions
• begin to comprehend decimals and how they are used in real life for money, taking temperatures and measuring accurately
• begin to divide two- and three-digit whole numbers by one-digit whole number
• know the multiplication table to times 10 (10 X 10)
• solve problems that require more than one operation using whole numbers.

• can understand and use numbers up to 100,000.
• investigate per cents and extend understanding of decimals and fractions
• begin adding and subtracting fractions with the same denominator.
• know the multiplication table to times 12 (12 X 10)
• solve problems of more than one operation (add, subtract, multiple and divide) with whole and decimal numbers

• can understand and use numbers up to 1 million
• begin to understand integers, such as -2, -1, 0, 1, 2
• use patterns to understand the divisibility rules of 2, 3, 4, 5, 6, 8, 9, and 10. For example, when even numbers are divided by 2, there is no remainder,
• Build operational skills include dividing a whole number by a fraction or a mixed number, (1 1/2)
• solve problems using more than one operation, involving whole numbers, decimals and fractions.

Base Ten Set: You kids may need a review of what place value. Use these blocks until the meaning of position is firmly understood before you move past 1,000.

Place Value Flip Chart: For younger grades, you can block off areas to the left that the kids do not need to know. On the other side is a flip chart for whole numbers and decimals.

Fraction Circles: Once you have taught the basic idea of fractions, you can use these circles to explore the relationships such as 1/3 = 2/6.

Fraction Tiles and Circles: Fractions can be applied to many different circumstances. The tiles and circles gives you the opportunity to demonstrate that circles are not the only circumstance where mathematicians use fractions. The tiles leads into parts of a group concept.

Fraction Towers: These towers are useful for both fractions and decimals. Fo younger grades you may want to limit the number of towers you use for introductory lessons.

Window Thermometer: Integers are hard for some kids to understand but when shown the everyday application of a thermometer, they begin to grasp the concept of negative numbers. The red and blue numbers help kids understand the concept of below zero.

Student Number Lines: These number lines will help your kids add and subtract when working with integers. Since they have 0 in the center, it is easy to show how to add negative numbers. The red and blue alternate numbers help with accuracy. ### Algebra

Although many parents think of algebra as part of an older child’s knowledge, many concepts can be taught at a younger grade by thinking of patterns in mathematics.

• the understanding of patterns is extended to classifying patterns as repeating or increasing
• work with algebraic statements for example, if n + 4 = 10, then n must be 6
• learn to write and read code to create geometric designs
• use the modeling process to analyze and create solutions for real-life situations, such as raising money through a read-a-thon

• continue classifying patterns as repeating, growing, and shrinking
• begin to create and solve algebraic equations with whole numbers, such as 6 + x = 20 – 4
• With coding, use their understanding of multiplication and ratios to create and execute code for patterns that grow
• use mathematical modeling to solve problems drawn from real-life, such as creating a design for a school wildlife area and calculating the cost of the trees, plants and benches

• continue exploration of patterns by starting to identify patterns that grow at a constant rate. For example, if someone drives 100 kilometers per hour, the distance travelled increases by 100 kilometers for each hour.
• solve algebraic expressions involving whole numbers and decimal tenths, and algebraic equations involving multiple terms, such as 3x + 2x = 10.
• use code to solve problems that involve optimization, such as finding the maximum area for a given perimeter
• use the process of mathematical modelling to solve problems drawn from real-life, such as finding several different ways to maximize areas in the school library and calculating the costs of each.

Jumbo, Magnetic Algebra Tiles: These tiles make algebra concepts visible for students. Some elements of this set can be introduced at the junior level and then used into the intermediate grades.

Foam Algebra Tiles: These are not magnetic but these tiles do make algebraic concepts visible to kids. Note that the red side is used for negative numbers.

### Data

• collect, organize, and display two or more data sets using frequency tables and multiple-bar graphs
• start to learn how to create an infographic, so that they can illustrate the meaning of the data.

• understand the importance of using various sampling techniques to get “good” data.
• create more sophisticated infographics
• learn how to identify when graphs are misleading.
• use experiments to understand the concept of probability.

• learn the difference between discrete data, such as the number of chairs in a room, and continuous data, such as the amount of time in seconds
• choose how to display these different types of data, including the use of broken-line graphs to show the change over time.
• learn different ways to describe probability. For example, there is a one in two chance of getting heads or tails, or there is a 50 percent chance of sun tomorrow.

Spinners (5): These different spinners can be used to collect data to be used for probability experiments. You can extend the experiment by making spinners that do not have equal parts to see if the probability changes.

Dice: You may have several sets of di from games you already have or you can order this colourful set. Many experiments with probability can be designed to check out your predictions.

### Spatial Sense

• understand the characteristics and properties of a rectangle
• determine the area of a rectangle
• understand the relationship between the various units in the metric system.

• understand the characteristics and properties of different types of triangles, including their angles and measurements
• continue using the metric system to measure length, area, mass and capacity, and learn how to convert from larger units to smaller ones.

• investigate the characteristics and properties of different four-sided shapes and find their areas
• measure angles with accuracy
• construct three-dimensional structures and learn to calculate surface area
• learn to convert from one unit to another in the metric system

Folding Geometric Shapes: This kit of shapes has the advantage of also including the nets (or patterns) for the shape. Some people have found the pieces a little difficult to use. The clear shapes also allow you to investigate the capacity of the various shapes. The nets help with learning about surface area.

Early Pattern Blocks: You have probably allowed your kids to make patterns when they were younger. Now these shapes can be examined for a more detailed investigation of their properties including surface area and measurement of angles.

Mini Geometric Shapes: These shapes are perfect for investigating the characteristics of various 3D solids. Use plasticine or sticky tack to count the corners (verticies), sides, and edges. Your kids can trace the various surfaces to design their own nets or calculate the surface area of various shapes.

Geometry Set: You will need the protractor to measure angles accurately. Take your time to teach how the 2 scales work, as this can be confusing for some kids. The triangular rulers are examples of 2 different triangles.

GeoStix: With these sticks and cards you can set up an exploration of triangles and rectangles. They can also be used for many different shapes as well.

### Financial Literacy

This area is a new section of the Ontario curriculum. Financial literacy is an essential life skill.

• explore the different ways to pay for goods and services
• investigate how consumers determine whether a purchase is good value for the price

• investigate different ways to transfer money between people and organizations, such as e-transfers and cheques
• calculate the total cost and change required for cash transactions for items priced in dollars and cents, using mental math and other strategies.
• calculate the best value for an item – for example, 4 drinks for \$3.60 compared to 3 drinks for \$2.40.
• design basic budgets and learn about the concepts of credit and debt.

• investigate the advantages and disadvantages of using different methods of payment for goods and services
• investigate different types of financial goals, identify and describe factors that could affect these goals, and outline steps to achieve them
• explain the concept of interest rates and identify interest rates and fees offered by banks and other financial institutions
• explore different ways to distribute resources: trading, lending, borrowing and donating

Buy It Right: In this game the players get to price, buy and sell items, and learn the value of money as they travel the game board. There are 3 levels of difficulty for more mature play. While this is strictly not a manipulative, it does set up a situation to practice money skills in a FUN environment.

Pay Day: In this game players need to find bargains and unexpected windfalls while holding on to hard-earned cash so that they will not be broke before Pay Day. This simulates the month to month struggle many families contend with.

### Social-Emotional

Layered over what students will know and do, are some expectations that address how students feel about their mathematical skills. These expectations also expand on how students problem solve using a wide variety of manipulatives and techniques to enhance their thinking.

It is a fact that many people struggle with mathematical thinking to solve daily problems. Through continual support of social-emotional growth for mathematical thinking, this negative attitude can be changed.

• continue learning about positive motivation. For example, students will use a variety of tools and strategies to try different non-standard units to measure the area of a tabletop, adjusting their techniques to reach a solution

• continue to develop healthy relationship skills while working with numbers. For example, play games that involve money, fractions, decimals, and whole numbers.
• develop positive interactions and patience as they take time to figure out the answer when it is their turn.

• continue to develop their sense of self. For example collect personal data such as the number of steps they take each day, minutes of screen time or how they feel after physical activity
• develop graphs and data visualization tools to provide information for reflective analysis

It is time to LEVEL UP your teaching of Mathematics.

Games and Manipulatives can do that for you.

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