Hands-on maths
Manipulatives are things you can touch and move to help you understand maths. Using your hands lets your brain see the idea before you work with numbers alone.
Popular manipulatives
| Tool | Helps with |
|---|---|
| Counters | counting, adding, sharing, odd and even |
| Linking cubes | counting, patterns, measuring, making trains |
| Base-ten blocks | place value, regrouping |
| Ten frames | numbers to 10, making ten |
| Pattern blocks | shapes, symmetry, fractions |
| Number lines | counting on, comparing, adding and subtracting |
| Play money | counting coins, making change |
| Geoboards | shapes, area, perimeter |
How to model a problem
To model a problem means to show it with objects or pictures.
Problem: 3 frogs sit on a log. 4 more jump on. How many frogs?
- Put out 3 counters for the first frogs.
- Add 4 more counters.
- Count them all: 7 frogs.
From objects to pictures to numbers
Learning often moves in three steps:
| Step | What you use | Example for 3 + 4 |
|---|---|---|
| Concrete | Real objects | 3 blocks and 4 blocks |
| Pictorial | Drawings | ●●● ●●●● |
| Abstract | Numbers and symbols | 3 + 4 = 7 |
When a problem feels hard, go back a step and draw it or build it.
Visual representations
A visual representation is a picture or diagram that shows a maths idea:
- Drawings of objects
- Number lines for jumps
- Ten frames for numbers up to 10
- Part-part-whole diagrams for addition and subtraction
- Arrays (rows and columns) for equal groups
Pattern blocks and shapes
Pattern blocks fit together: two red trapezoids make a yellow hexagon, and three green triangles make a red trapezoid. This shows halves and thirds before you learn fraction symbols.
Worked examples
Example 1: Use counters to show 8 − 3.
Put out 8 counters, take away 3, and count what’s left: 5.
Example 2: Use base-ten blocks to show 24.
2 rods (tens) and 4 cubes (ones).
Example 3: Draw a picture for "4 children each have 2 apples."
4 groups of ●●. Count: 8 apples.
Common mistakes
- Playing instead of modelling. Use the tools to match the problem.
- Not connecting the model to the numbers. Write the number sentence after you build it.
- Using messy arrangements that make counting hard. Line things up.
Try it yourself
- Which manipulative would you use to show 36?
- Model 5 + 2 with a drawing.
- How many green triangles make a red trapezoid?
- What does "concrete" mean in the three steps?
Answers: 1) base-ten blocks 2) ●●●●● ●● = 7 3) 3 4) using real objects
Quick recap
- Manipulatives like counters, base-ten blocks and pattern blocks help you model maths.
- Move from objects to pictures to numbers.
- A visual representation makes an idea easier to see.