Mathematics
Advanced
50 mins
Teacher/Student led
+80 XP
What you need:
IWB/Projector/Large Screen

Properties of Operations and the Distributive Law

Learn why splitting a factor makes multiplication easier by using the area model. Discover the distributive law: a × (b + c) = a × b + a × c, and test whether it works for subtraction too.

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    1 - Getting Started ~4 mins

    Here is a question to crack in your head: what is 6 × 13?

    Tip

    Don't reach for the column method yet. Could you split that 13 into 10 and 3 to make the multiplying easier? Have a go, then we'll find out together why that splitting trick always works.

    2 - Watch and Notice ~11 mins

    Illustration for Watch and Notice

    6 × 13 = 6 × 10 + 6 × 3

    Let's look at these together. Here is 6 × 13 drawn as a rectangle, 6 down and 13 across. A factor is one of the two numbers we are multiplying. Here we have split one factor, the 13, into a 10 piece and a 3 piece. What do you notice about the two smaller rectangles?

    4 × 25 = 4 × 20 + 4 × 5

    Same idea, different numbers. The 25 side splits into 20 and 5. Read off each smaller rectangle's area, then add them.

    9 × 16 = 9 × 10 + 9 × 6

    Once more, splitting at the tens. The 16 side splits into 10 and 6. Watch how the two rectangles tile together to make the one whole rectangle, with no gaps and no overlaps.

    The rule we keep seeing

    Every time, multiplying by a sum gives the same answer as multiplying each part and adding. In short: a × (b + c) = a × b + a × c. The brackets just mean do the adding inside first.

    3 - Try It Together ~8 mins

    Now we work one product together on the area model: 8 × 14. We'll split one factor (one of the two numbers we are multiplying), the 14, into 10 and 4, fill in each smaller rectangle, and add the two parts to check the whole.

    Tip

    Where would you cut the 14 to make the multiplying easy? Let's find out together.

    Split the product: 8 × 14

    4 - Write the Splits in Your Copy ~5 mins

    COPYBOOK MOMENT

    In your maths copy, write these three products and beside each show the distributive split, then work out both sides:

    • 5 × 23 = 5 × 20 + 5 × 3
    • 9 × 16 = 9 × 10 + 9 × 6
    • 6 × 17 = 6 × 10 + 6 × 7

    Work out both sides of each line and underline the two equal totals.

    5 - Class Challenge ~9 mins

    Today we work through these splits together, getting trickier each time: 7 × 13, then 6 × 24, then 8 × 35, then 9 × 27. Build each on the area model, fill both rectangles, and add the parts.

    Split and find the product

    6 - What Did We Notice? ~4 mins

    One pupil says the splitting trick also works for subtraction, so 6 × 18 should equal 6 × 206 × 2. Another pupil isn't sure. Look at the picture below: we start with the big 6 × 20 rectangle and cut off the extra 6 × 2 strip on the end.

    What is left after we cut the strip off? So who is right, and how does the picture settle it without doing the full sum first?

    7 - What's Next ~3 mins

    Today you learned that multiplying by a sum is the same as multiplying each part and adding the answers, and you saw why the area picture makes that true every time, for both plus and minus.

    Next we'll meet letter-symbols: using a letter to stand for a number we don't know yet, the first step into writing things in algebra.

    Pupil practice
    Module 10 · Algebra: Patterns, Expressions and Equations Mixed
    Lesson 106 · Properties of Operations and the Distributive Law
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