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Illustrative teaching example · Fractions
Different pieces. One common size.
A common wrong turn⅓ + ¼ = ²⁄₇?
Adding the tops and the bottoms changes the size of the pieces. A third and a quarter need a common unit before we can add them.
Make the pieces the same size.
One third is four twelfths. One quarter is three twelfths. We have seven pieces, each one twelfth of the same whole.
⅓ = ⁴⁄₁₂
¼ = ³⁄₁₂
⁴⁄₁₂ + ³⁄₁₂ = ⁷⁄₁₂
Change the units. Keep the value.
Twelve is a multiple of both 3 and 4. Multiplying the top and bottom by the same number keeps a fraction’s value.
Rewrite one third⅓ × ⁴⁄₄ = ⁴⁄₁₂
Rewrite one quarter¼ × ³⁄₃ = ³⁄₁₂
Add the twelfths⁴⁄₁₂ + ³⁄₁₂ = ⁷⁄₁₂
A fresh question
What is ½ + ⅕?
One practice question is a chance to use the idea. It is not a placement result or an enrollment mastery check.
Illustrative teaching example · Negative numbers
A direction, not just a sign.
A common wrong turn−3 + 8 = −11?
The negative sign tells us where we start. Adding a positive number moves us toward larger numbers; it does not make the answer more negative.
Start at −3. Move eight places right.
Three steps bring us to zero. Five more steps bring us to 5.
3 steps to zero5 more steps
−3−2−1012345
−3 + 8 = 5
Split eight into three and five.
Negative three and positive three cancel. The remaining five stays positive.
Split the positive amount−3 + (3 + 5)
Group the zero pair(−3 + 3) + 5
See what remains0 + 5 = 5
A fresh question
What is −5 + 9?
One practice question is a chance to use the idea. It is not a placement result or an enrollment mastery check.
Illustrative teaching example · Simple equations
Keep both sides in balance.
A common wrong turn4x + 2 = 22 → x = 5½?
Dividing 22 by 4 before accounting for the extra 2 skips part of the equation. Undo the added 2 on both sides first.
Four equal groups, plus two, make twenty-two.
Take two away from both sides. The four equal groups now share twenty, so each group is five.
xxxx+ 2= 22
xxxx= 20
x= 5
Undo addition before multiplication.
Whatever we do to one side, we also do to the other. Then check the result in the original equation.
Subtract two from both sides4x = 20
Divide both sides by fourx = 5
Check in the original4 × 5 + 2 = 22
A fresh question
If 3x + 4 = 25, what is x?
One practice question is a chance to use the idea. It is not a placement result or an enrollment mastery check.
Understanding comes before moving on.
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