Expression Lab

What is an expression?

An expression is a recipe written with numbers and letters. 3x + 5 says "take some number x, triple it, then add 5." There is no equals sign, so nothing is being solved yet. It is just a description of a calculation.

The letter is a variable. It stands for a number you might not know yet, or for any number you want to try.

What are exponents?

An exponent is a short way to write repeated multiplication. 3⁴ means 3 × 3 × 3 × 3, which is 81. The little 4 says how many copies of the 3 to multiply.

A common mistake is to multiply 3 × 4 and get 12. The exponent says the base is used as a factor, not multiplied once.

The parts of an expression

Take 3x + 5. It has two terms, 3x and 5, separated by the plus sign.

The 3 is the coefficient. It is the number multiplying the x. The 5 is the constant, because it is just a number and does not change when x changes.

If your kid can point at the coefficient and the constant, they know the vocabulary.

Substituting a value

Substituting means putting a number in for the letter. If x is 4, then 3x + 5 becomes 3 × 4 + 5, which is 17.

This is also how formulas work. V = s³ finds the volume of a cube with side s. If s is 3, the volume is 27.

Equivalent expressions

Two expressions are equivalent when they give the same answer for every number you try. y + y + y and 3y are equivalent. Whatever y is, both give three times it.

The distributive property makes equivalent expressions. 3(x + 2) means three groups of (x + 2). That is three x's and six 1s, so 3(x + 2) = 3x + 6. The area model in the module shows exactly this as a rectangle of tiles.

What they're learning

• An expression with a letter describes a calculation, not one fixed answer.
• An exponent is repeated multiplication.
• Each part of an expression has a name: term, coefficient, constant.
• Substituting a value turns an expression into a number.
• Equivalent expressions give the same value for every number you try.

Aliases

• Algebraic expressions
• Simplifying expressions
• Combining like terms
• Distributive property
• Substitution

Refresher

An expression is a calculation with numbers and letters and no equals sign, like 3x + 5. The number multiplying a letter is the coefficient. A number on its own is the constant. Substituting means putting a value in for the letter, and equivalent expressions give the same answer for every value.

Try this together

Two or three minutes at the table is enough.

1. Open Exponents. Set the base to 3 and the exponent to 4. Ask your kid to say it as repeated factors before reading the answer.
2. Switch to Name the parts and look at 3x + 5. Ask which number is the coefficient and which is the constant.
3. Switch to Substitute. Set x to 4 and work out 3x + 5 together.
4. Switch to Equivalent. Point at the rectangle and ask how many x-tiles and how many 1-tiles it holds.

If your kid says 3x + 5 is 8 when x is 1, that is right for that one value. The point of the letters is that the expression works for every value.

Words you might hear

• Expression: a calculation with numbers and letters, with no equals sign.
• Variable: a letter standing for a number, like x.
• Term: a piece separated by + or −. 3x + 5 has two terms.
• Coefficient: the number multiplying a variable, like the 3 in 3x.
• Constant: a term that is just a number, like the 5 in 3x + 5.
• Equivalent: two expressions that give the same value for every number.

Deeper dive

Read 3x + 5 out loud as "three x's plus five". If x is 4, that is 4 + 4 + 4 + 5, which is 17. The letter is a label, not a mystery.

An exponent is repeated multiplication. 3⁴ means 3 × 3 × 3 × 3 = 81, not 3 × 4. The distributive property turns 3(x + 2) into 3x + 6, because it is three groups of x plus three groups of 2. The rectangle in the module is that same statement built out of tiles.